CO-ORDINATE GEOMETRYeinsteinclasses.com/JEE Main Website/MATHEMATICS/Co...its equation, translation of axes, slope of a line, parallel and perpendicular lines, intercepts of a line

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road

New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 1

CO-ORDINATE GEOMETRY

Sy l labus :

Cartesian system of rectangular co-ordinates in aplane; distance formula, section formula, locus andits equation, translation of axes, slope of a line,parallel and perpendicular lines, intercepts of a lineon the coordinate axes.

Straight lines

Various forms of equations of a line, intersection oflines, angles between two lines, conditions forconcurrence of centroid, orthocentre andcircumcentre of a triangle, equation of family oflines passing through the point of intersection oftwo lines.

Circles, conic sections

Standard form of equation of a circle, general formof the equation of a circle, its radius and centre,equation of a circle when the end point of adiameter are given, points of intersection of a lineand a circle with the centre at the origin andcondition for a line to be tangent to a circle,equation of the tangent. Sections of cones,equation of conic sections (parabola, ellipse andhyperbola) in standard forms, condition fory = mx + c to be tangent and point (s) of tangency.


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MCG – 2

CONCEPTS (Straight Line)

C1 Co-ordinate axes : X-axis and Y-axis together are

called axes of co-ordinates.

OA = PB = x (is known as abscissa)

AB = OP = y (is known as ordinate)

C2 Distance Formula : In Cartesian co-ordinate the distance between the points P(x1, y

1) and Q (x

2, y

2) is

given by

PQ = 221

221 )yy()xx(

C3 Condition of Collinearity using the distance formula : For three points to be collinear, prove that thesum of the distances between two points is equal to the third pair of points.

C4 Section Formula :

1. Internal Division : If (x, y) are the coordinates of the point that divides the line segment joining the points

(x1, y

1) and (x

2, y

2) internally in the ratio m : n, then

nm

nymyyand

nm

nxmxx 1212

.

2. External Division : If (x, y) are the coordinates of the point that divides the line segment joining the points

(x1, y

1) and (x

2, y

2) externally in the ratio m : n, then

nm

nymyyand

nm

nxmxx 1212

.

Practice Problems :

1. The line x + y = 4 divides the line joining the points (–1, 1) and (5, 7) in the ratio

(a) 2 : 3 (b) 1 : 2 (c) 1 : 1 (d) 4 : 3

[Answers : (1) b]

C5 Centroid or centre of gravity Triangle : The centroid of a triangle is the point of intersection of itsmedians. The centroid divides the medians in the ratio 2 : 1. If the coordinate of a triangle is (x

1y

1) (x

2y

2)

(x3y

3) then its centroid is

3

yyy,

3

xxx 321321 .

Incentre of a Triangle : The point of intersection of the internal bisectors of the angles of a triangle is theincentre.

Circumcentre : The circumcentre of a triangle is the point of intersection of the perpendicular bisectors ofthe sides of a triangle.

Orthocentre : The orcthocentre of a triangle is the point of intersection of altitudes drawn from vertex ofthe triangle to the opposite sides.

Practice Problems :

1. If the orthocentre and centroid of a triangle are (–3, 5) and (3, 3), then its circumcentre is

(a) (6, 2) (b) (3, –1) (c) (–3, 1) (d) (–3, 5)

2. The orthocentre of the triangle formed by the lines xy = 0 and x + y = 1 is

(a)

2

1,

2

1(b)

3

1,

3

1(c) (0, 0) (d)

4

1,

4

1


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MCG – 3

3. If the vertices P, Q, R are rational points. A rational point is a point both of whose co-ordinates arerational numbers. Which of the following points of the triangle PQR is always rational point

(a) centroid (b) orthocentre (c) circumcentre (d) All of these

4. Vertices of a ABC are A (2, 2), B (–4, –4), C (5, –8). Then length of the median through C is

(a) 65 (b) 117 (c) 85 (d) 113

5. Let PS be the median of the triangle with vertices P(2, 2), Q(6, –1) and R(7, 3). The equation of theline passing through (1, –1) and parallel to PS is

(a) 2x – 9y – 7 = 0 (b) 2x – 9y – 11 = 0

(c) 2x + 9y – 11 = 0 (d) 2x + 9y + 7 = 0

6. The orthocentre of the triangle with vertices

2

1,2and

2

1,

2

1,

2

)13(,2 is

(a)

6

33,

2

3(b)

2

1,2 (c)

4

23,

4

5(d)

2

1,

2

1

[Answers : (1) a (2) c (3) d (4) c (5) d (6) b]

C6 Area of a Triangle : Area of the triangle with vertices (x1, y

1), (x

2, y

2) and (x

3, y

3) is given by

1yx

1yx

1yx

2

1

33

22

11

= |)}yy(x)yy(x)yy(x{|2

1213132321 .

Practice Problems :

1. P (3, 1), Q (6, 5) and R (x, y) are three points such that the angle PRQ is a right angle and the area of R Q P = 7, then the number of such points R is

(a) 0 (b) 1 (c) 2 (d) 4

2. The area of a triangle is 5 and two of its vertices are A (2, 1), B (3, –2). The third vertex which lies online y = x + 3 is

(a)

2

13,

2

7(b)

2

5,

2

5

(c)

2

3,

2

3(d) both (a) and (c) are correct

[Answers : (1) c (2) d]

C7 Locus : When a point moves so as always to satisfy a given condition, or conditions, the path it trace out iscalled its Locus under these conditions.

Practice Problems :

1. Locus of centroid of the triangle whose vertices are (a cos t, a sin t), (b sin t, – b cos t) and (1, 0), wheret is a parameter, is

(a) (3x – 1)2 + (3y)2 = a2 + b2 (b) (3x + 1)2 + (3y)2 = a2 + b2

(c) (3x + 1)2 + (3y)2 = a2 – b2 (d) (3x – 1)2 + (3y)2 = a2 – b2


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MCG – 4

2. Locus of a point which moves such that sum of the squares of its distances from the sides of a squareof side unity is 9, is

(a) Straight line (b) Circle (c) Parabola (d) None

[Answers : (1) a (2) b]

C8 STRAIGHT LINE : The equation be of the first degree (i.e. if it contain no products, squares, or higherpowers or x and y) then locus corresponding is always a straight line.

Slope of a Straight Line : Let a straight line makes an angle /2 and 0 < , 2

with the positive

direction of x-axis. Then tan is called the slope of the line. It is denoted by m.

Slope of Straight Line Joining Two Points :

Let A(x1, y

1) and B(x

2, y

2) be two points. Then the slope of AB (tan ) is

12

12

xx

yy

if x

1 x

2.

C9 Condition of collinearity of three points A(x1, y

1), B(x

2, y

2) and C(x

3, y

3) using the concept of slope.

slope of AB = slope of BC = slope of AC

C10 Angle Between Two Straight Lines : Let be the acute angle between two straight lines whose slopes are

m1 and m

2 respectively and m

1m

2 –1, then

21

21

mm1

mmtan

.

Practice Problems :

1. The lines x cos + y sin = p1 and x cos + y sin = p

2 will be perpendicular if

(a) = (b)2

(c)

2

(d)

2

[Answers : (1) b]

C11 A Point and a Straight Line :

(i) Perpendicular distance of a point (x1, y

1) from a straight ax + by + c = 0 is

22

11

ba

cbyax

.

(ii) Coordinates of the foot of perpendicular drawn from a point (x1, y

1) to the line ax + by + c = 0 are given by

221111

ba

cbyax

b

yy

a

xx

.

Practice Problems :

1. The foot of the perpendicular from point (2, 4) upon x + y = 1 is

(a)

2

3,

2

1(b)

2

3,

2

1(c)

2

1,

3

4(d)

2

1,

4

3

[Answers : (1) b]


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MCG – 5

C12 Distance between Parallel Lines : Distance parallel lines ax + by + c1 = 0 and ax + by + c

2 = 0 is equal to

22

21

ba

cc

.

C13 Reflection of a point about a line : Coordinates of the image of the point (x1, y

1) in the line ax + by + c =

0 are given by 22

1111

ba

cbyax2

b

yy

a

xx

.

Practice Problems :

1. The reflection of the point (4, –13) in the line 5x + y + 6 = 0 is

(a) (–1, –14) (b) (3, 4) (c) (1, 2) (d) (–4, 13)

[Answers : (1) a]

C14 Family of Straight Lines :

Let L1 a

1x + b

1y + c

1 = 0 and L

2 a

2x + b

2y + c

2 = 0 be two non-parallel lines. Then the equation of the

family of straight line passing through the intersection of the lines L1 = 0 and L

2 = 0 is L

1 + L

2 a

1x + b

1y

+ c1 + (a

2x + b

2y + c

2) = 0.

C15 Concurrent lines : The three lines are concurrent if they meet in a point.

Three lines a1x + b

1y + c

1 = 0, a

2x + b

2y + c

2 = 0 & a

3x + b

3y + c

3 = 0 are concurrent if

0

cba

cba

cba

333

222

111

Practice Problems :

1. If the lines x + a y + a = 0, b x + y + b = 0 and c x + c y + 1 = 0 (a, b and c being distinct 1) are

concurrent, then the value of 1c

c

1b

b

1a

a

is

(a) –1 (b) 0 (c) 1 (d) none of these

2. If the lines x = a + m, y = –2 and y = mx are concurrent, the least value of [a] is

(a) 0 (b) 2 (c) 22 (d) none of these

3. The equation of the line with slope 2

3 which is concurrent with the lines 4x + 3y – 7 = 0 and

8x + 5y – 1 = 0 is

(a) 3x + 2y – 2 = 0 (b) 3x + 2y – 63 = 0

(c) 2y – 3x – 2 = 0 (d) none of the above

4. Three lines px + qy + r = 0, qx + ry + p = 0 and rx + py + q = 0 are concurrent if

(a) p + q + r = 0 (b) p2 + q2 + r2 = pq + qr + rp

(c) p3 + q3 + r3 = 3pqr (d) both (a) and (c) are correct

[Answers : (1) c (2) c (3) a (4) d]

C16 Pair of Straight Lines : ax2 + 2hxy + by2 = 0 represents a pair of straight lines passing through the origin.If their slopes are m

1 and m

2, then m

1 + m

2 = –2h/b and m

1m

2 = a/b.


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MCG – 6

Practice Problems :

1. The equation ax2 + hxy + by2 = 0 represents a pair of real and different straight lines through theorigin if

(a) h2 = 4ab (b) h2 < 4ab (c) g2 > 4ab (d) none of these

[Answers : (1) c]

C17 Angle Between Pair of Straight Lines : Let be the acute angle between pair of straight lines ax2 + 2hxy

+ by2 = 0 then |ba|

abh2tan

2

.

Practice Problems :

1. If the angle between the two lines represented by 2x2 + 5xy + 3y2 + 6x + 7y + 4 = 0 is tan–1 (m), thenm is equal to

(a) 1/5 (b) –1 (c) –2/3 (d) none of these

2. The difference of the tangents of the angles which the lines x2(sec2 – sin2) – 2xy tan + y2sin2 = 0make with x-axis is

(a) 2 tan (b) 2 (c) 2 cot (d) sin 2

[Answers : (1) a (2) b]

C18 Equation of the Bisectors of the angles between the lines : Equation of the straight lines bisecting the

angles between the pair of straight lines ax2 + 2hxy + by2 = 0 is h

xy

ba

yx 22

.

Practice Problems :

1. If the pair of straight lines x2 – 2pxy – y2 = 0 and x2 – 2qxy – y2 = 0 be such that each pair bisects theangle between the other pair, then

(a) p = –q (b) pq = 1 (c) pq = –1 (d) p = q

[Answers : (1) c]

C19 General Equation of Second Degree : General equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 representstwo straight lines if abc + 2fgh – af2 – bg2 – ch2 = 0.

Practice Problems :

1. If the equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a pair of parallel lines, then the distancebetween them is

(a) 22

2

ah

acg

(b)

)ba(a

acg2

(c) 22

2

ah

acg2

(d)

)ba(a

acg2

2

[Answers : (1) c]

C20 Time saving tips :

1. Three points (x1, y

1), (x

2, y

2) and (x

3, y

3) are collinear

(i) If 3

3

2

2

1

1

x

y

x

y

x

y , then these points lie on a straight line passing through the

origin.

(ii) If 3

3

2

2

1

1

x

y

x

y

x

y is not true but

13

13

12

12

xx

yy

xx

yy

, then the points lie on a straight


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MCG – 7

line not passing through the origin.

2. OAB is obtuse angled if the points A and B lie in opposite quadrants.

3. If a1x + b

1y + c

1 = 0; a

2x + b

2y + c

2 = 0 and a

3x + b

3y + c

3 = 0 are the sides of a triangle, then the area of the

triangle is given by (without solving the vertices).

333

222

111

321 cba

cba

cba

CCC2

1

when C1, C

2, C

3 are the co-factors of c

1, c

2, c

3 in the determinant

i.e., C1 = a

2b

3 – a

2b

2, C

2 = a

3b

1 – a

2b

3 and C

3 = a

1b

2 – a

2b

1.

4. Reflection of the point (, ) about the line y = x is (, ).

5. The line joining the points (x1, y

1) and (x

1, y

2) is divided by the X-axis in the ratio

2

1

y

y and by Y-axis in

the ratio

2

1

x

x .

6. If (x1, y

1), (x

2, y

2) and (x

3, y

3) be respectively the vertices A, B, C of a triangle, then angle A is acute or

obtuse according as (x1 – x

2) (x

1 – x

3) + (y

1 – y

2) (y

1 – y

3) is positive or negative.

7. Area of the triangle formed by the lines of the form y = m1x + c

1, y = m

2x + c

2 and y = m

3x + c

3 is

21

221

13

213

32

232

mm

)cc(

mm

)cc(

mm

)cc(

2

1.

8. The area of a rohmbus with sides ax ± by ± c = 0 is ab

c2 2

.

9. The area of a parallelogram formed by the lines a1x + b

1y + c

1 = 0, a

1x + b

1y + d

1 = 0, a

2x + b

2y + c

2 = 0, and

a2x + b

2y + d

2 = 0 is

1221

2211

baba

)dc)(dc(

.

10. If the lines represented by a1x2 + 2h

1xy + b

1y2 = 0 and a

2x2 + 2h

2xy + b

2y2 = 0 are equally inclined to each

other then 2

22

1

11

h

ba

h

ba

11. Centroid of the triangle obtained by joining the middle points of the sides of a triangle is the same as thecentroid of the original triangle.

12. If two vertices of an equilateral triangle are (x1, y

1) and (x

2, y

2) then co-ordinates of the third vertex are

2

)xx(3yy;

2

)yy(3xx 21212121 . The vertices of the equilateral triangle do not have

integral coordinates.

13. Centroid, orthocentre, circumcentre coincides for the equilateral triangle.

14. Shifting of the origin does not alter the area of triangle.

15. The orthocentre of a triangle having vertices (, ), (, ) and (, ) is (, ).

e.g., the orthocentre of the triangle having vertices (4, 5), (5, 4) and (4, 4) is (4, 4).


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MCG – 8

16. The image of the line ax + by + c = 0 about y-axis –ax + by + c = 0.

17. If the circumcentre and centroid of a triangle are respectively (, ), (, ) then orthocentre will be(3 – 2, 3 – 2).

18. The length of the intercept cut by the lines represented by ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 on x-axis is

|a|

acg2 2 .

19. A line passing through (x1, y

1) and parallel to ax + by + c = 0 is a (x – x

1) + b (y – y

1) = 0.

20. Circumcentre of the triangle formed by the origin (0, 0) and the points (x1, y

1) and (x

2, y

2) is

222

22

121

21

322

222

21

211

2

22

11

1

1

3

1

2

x2)yx(

x2)yx(;

)yx(y2

)yx(y2;

y2x2

y2x2where,

.

21. Circumcentre of the triangle formed by the points (x1, y

1), (x

2, y

2) and (x

3, y

3) is the same as that of the

triangle formed by the points (0, 0), (x2 – x

1, y

2 – y

1), (x

3 – x

1, y

3 – y

1).

22. Orthocentre of the triangle formed by going the points (0, 0), (x1, y

1), (x

2, y

2) is given by [q (y

2 – y

1), – q (x

2

– x1)] where

1221

2121

yxyx

yyxxq

.

23. Orthocentre of the triangle formed by the points

,

1is

1,,

1,,

1, .

24. The product of perpendicular drawn from any point (, ) on the lines ax2 + 2hxy + by2 = 0 is

22

22

h4)ba(

bh2a

.


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MCG – 9

CONCEPTS (Circle)

A circle is the locus of a point moving in a plane so that it always remains at a constant distance from a fixedpoint. The fixed point is called its center and the constant distance is called its radius.

C1 Equation of a Circle in different forms :

Forms Equation Figure

Standard Form (x – h)2 + (y – k)2 = r2

here (h, k) is the centre andr is the radius

General Formx2 + y2 + 2gx + 2fy + c = 0 is the circle whose center is at (–g, –f)

and radius is cfg 22 .

Notes : (i) If g2 + f2 – c > 0, then the circle iscalled a real circle.

(ii) If g2 + f2 – c = 0, then the circle iscalled a point circle.

(iii) If g2 + f2 – c < 0, then the circle iscalled an imaginary circle.

(iv) The lengths of intercepts made bythe circle x2 + y2 + 2gx + 2fy + c = 0with x and y axes are

cf2andcg2 22

respectively.

Diameter Form If the end points of the diameter of a circle are (x1y

1) and (x

2y

2)

then equation of the circle is(x – x

1) (x – x

2) + (y – y

1) (y – y

2) = 0


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MCG – 10

C2 Straight Line and a Circle :

Let L = 0 be a line & S = 0 be a circle. If r is the radius of the circle & p is the length of the perpendicularfrom the centre on the line, then

(i) p > r the line does not meet the circle i.e. passes out side the circle.

(ii) p = r the line touches the circle. (It is tangent to the circle)

(iii) p < r the line is a sacant of the circle

(iv) p = 0 the line is a diameter of the circle

Note the following points :

The line y = mx + c touches circle x2 + y2 = a2 if c2 = a2(1 + m2) at point

c

a,

c

ma 22

.

Practice Problems :

1. If the straight line y = mx is outside the circle x2 + y2 – 20 y + 90 = 0 then

(a) m > 3 (b) m < 3 (c) |m| > 3 (d) |m| < 3

[Answers : (1) d]

C3 The length of the intercept cut off from a line by a circle :

The intercepts made by the circle x2 + y2 + 2gx + 2fy + c = 0 on the co-ordinates axes are cf2&cg2 22

respectively. If

g2 = c circle touches the x axis

Similarly,

f2 = c circle touches the y-axis

C4 Tangent to a Circle :

Point Form :

(a) Equation of the tangent to the circle S x2 + y2 + 2gx + 2fy + c = 0 at the point(x

1, y

1) is T xx

1 + yy

1 + g(x + x

1) + f (y + y

1) + c = 0.

(b) The equation of the tangent to the circle x2 + y2 = r2 if c2 = r2 (1 + m2). Hence, equation of tangent

is 2m1rmxy and the point of contact is

c

r,

c

mr 22

.

Practice Problems :

1. The tangent to x2 + y2 = 9 which is parallel to y-axis and does not lie in the first quadrant touches, thecircle at the point

(a) (3, 0) (b) (–3, 0) (c) (0, 3) (d) (0, –3)

2. If the line x sin – y cos = a touches the circle x2 + y2 = a2, then

(a)22

(b) 0 (c) – (d) R

3. If x cos + y sin = p is a tangent to the circle x2 + y2 = 2q(x cos + y sin ) then the set of possiblevalues of p is

(a) {q} (b) {0, q} (c) {0, 2q} (d) {q, 2q}

[Answers : (1) a (2) d (3) c]


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MCG – 11

C5 Normal : If a line is normal/orthogonal to a circle then it must pass through the centre of the circle. Using

this fact normal to the circle x2 + y2 + 2gx + 2fy + c = 0 at (x1, y

1) is )xx(

gx

fyyy 1

1

11

.

Practice Problems :

1. To which of the following circles the line y – x + 3 = 0 is normal at the point

2

3,

2

33 ?

(a) 92

3y

2

33x

22

(b) 0

2

3y

2

3x

22

(c) x2 + (y – 3)2 = 9 (d) (x – 3)2 + y2 = 9

[Answers : (1) c]

C6 Length of Tangent : The length of the tangent from the point P(x1, y

1) to the circle

x2 + y2 + 2gx + 2fy + c = 0 is equal to

11121

21 Scfy2gx2yx

Note : If PQ is a length of the tangent from a point P to a given circle, then PQ2 is called the power of thepoint with respect to a given circle.

Practice Problems :

1. If the square of the length of the tangents from a point P to the circle x2 + y2 = a2, x2 + y2 = b2 andx2 + y2 = c2 are in A.P. Then

(a) a, b, c are in A.P. (b) a, b, c are in A.P.

(c) a2, b2, c2 are in A.P. (d) a2, b2, c2 are in G.P.

[Answers : (1) c]

C7 Combined Equation of Pair of Tangents : Combined equation of the pair of tangents to the circle S x2

+ y2 + 2gx + 2fy + c = 0 drawn from the point (x1, y

1) is T2 = SS

1, where T and S

1 have their usual meaning.

Practice Problems :

1. The angle between a pair of tangents drawn from a point T to the circle

x2 + y2 + 4x – 6y + 9 sin2 + 13 cos2 = 0 is 2. The equation of the locus of the point T is

(a) x2 + y2 + 4x – 6y + 4 = 0 (b) x2 + y2 + 4x – 6y – 9 = 0

(c) x2 + y2 + 4x – 6y – 4 = 0 (d) x2 + y2 + 4x – 6y + 9 = 0

[Answers : (1) d]

C8 Chord of Contact : Let the circle is S x2 + y2 + 2gx + 2fy + c = 0, then the equation of the chord of contactis given by T = 0.

xx1 + yy

1 + g(x + x

1) + f(y + y

1) + c = 0

Practice Problems :

1. If the chord of contact of tangents drawn from a point on the circle x2 + y2 = a2 to the circlex2 + y2 = b2 touches the circle x2 + y2 = c2, then a, b, c are in

(a) A.P. (b) G.P. (c) H.P. (d) none of these

[Answers : (1) b]


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 12

C9 Equation of a Chord in Terms of its Mid Point : Let (x1, y

1) be the mid point of a chord of the circle S

x2 + y

2 + 2gx + 2fy + c = 0, then the equation of the chord is T = S

1.

C10 Director Circle : The locus of the point of intersection of two perpendicular tangents to a circle is calledthe Director Circle.

Practice Problems :

1. P is the point of intersection of two perpendicular tangents to the circle x2 + y2 = a2. Locus of P is

(a) x2 + y2 = 3a2 (b) x2 + y2 = 2a2 (c) x2 + y2 = 4a2 (d) none of these

[Answers : (1) b]

C11 Family of Circles :

(i) Let S x2 + y2 + 2gx + 2fy + c = 0 be a circle and L ax + by + k = 0 be a line intersecting S = 0, then theequation of the family of circles passing through the intersection of the given circle and the line is S + L= 0, where is a parameter.

(ii) Let S = 0 and S’ = 0 be two intersecting circles. Then equation of the family of circles passing throughpoints of intersection of S = 0 and S’ = 0 is S + S’ = 0, where – 1.

C12 Two Circles : Let there be two circles with centers at C1 and C

2 and radii r

1 and r

2 respectively, then

(i) Two circles are exterior to each other if r1 + r

2 < C

1C

2

(ii) Two circles touch each other externally if r1 + r

2 = C

1C

2

(iii) Two circles intersect each other in two points if |r1 – r

2| < C

1C

2 < r

1 + r

2

(iv) Two circles touch each other internally if |r1 – r

2| = C

1C

2

(v) One circle is interior to the other if C1C

2 < |r

1 – r

2|.

Practice Problems :

1. The two circles x2 + y2 +ax = 0 and x2 + y2 = c2 (with c > 0) touch each other if

(a) c = |a| (b) 2a = |c| (c) 2c = a (d) none of these

2. If one of the circles x2 + y2 + 2ax + c = 0 and x2 + y2 + 2bx + c = 0 lies with in the other, then

(a) ab > 0, c > 0 (b) ab > 0, c < 0 (c) ab < 0, c > 0 (d) none of these

3. The circles whose equations are x2 + y2 + c2 = 2ax and x2 + y2 + c2 – 2by = 0 will touch one anotherexternally if

(a) 222 a

1

c

1

b

1 (b) 222 b

1

a

1

c

1

(c) 222 c

1

b

1

a

1 (d) none of these

4. If two circles a(x2 + y2) + bx + cy = 0 and A(x2 + y2) + Bx + Cy = 0 touch each other, then

(a) aC = cA (b) bC = cB (c) aB = bA (d) aA = bB = cC

5. The locus of the centre of the circles which touch both the circles x2 + y2 = a2 and x2 + y2 = 4axexternally has the equation

(a) 12 (x – a)2 – 4y2 = 3a2 (b) 9 (x – a)2 – 5y2 = 2a2

(c) 8x2 – 3(y – a)2 = 9a2 (d) none of these

6. Two circles (x – 1)2 + (y – 3)2 = r2 and x2 + y2 – 8x + 2y + 8 = 0 intersect in two distinct points if

(a) 2 < r < 8 (b) r < 2 (c) r = 2 (d) r > 3

[Answers : (1) a (2) a (3) c (4) b (5) a (6) a]


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 13

C13 Common Tangents to two Circles :

Case Number of Tangents Condition

(i) 4 common tangents r1 + r

2 < c

1 c

2

(2 direct and 2 transverse)

(ii) 3 common tangents r1 + r

2 = c

1 c

2

(iii) 2 common tangents |r1 – r

2| < c

1 c

2 < r

1 + r

2

(iv) 1 common tangent |r1 – r

2| = c

1 c

2

(v) No common tangent c1 c

2 < |r

1 – r

2|

(Here c1c

2 is distance between centres of two circles)

Practice Problems :

1. The number of common tangents to the circles x2 + y2 = 4 and x2 + y2 – 6x – 8y = 24 is

(a) 0 (b) 1

(c) 3 (d) 4

[Answrs : (1) b]

C14 Angle of Intersection of Two Circles : Let be the angle of intersection of two circles whose centers are

at C1 and C

2 and their radii are r

1 and r

2 respectively, then

21

221

22

21

rr2

CCrrcos

.

Notes :

(i) If = 900, then the circles are said to be Orthogonal and then 2

212

22

1 CCrr .

(ii) Circles x2 + y2 + 2gx + 2fy + c = 0 and 0cyf2xg2yx 22 are orthogonal ifff

ccff2gg2 .

Practice Problems :

1. The circles x2 + y2 + x + y = 0 and x2 + y2 + x – y = 0 intersect at an angle of

(a) /6 (b) /4 (c) /3 (d) /2

2. If the circles x2 + y2 + 2x + 2ky + 6 = 0 and x2 + y2 + 2ky + k = 0, intersect orthogonally then k is

(a) 2 or –3/2 (b) – 2 or – 3/2 (c) 2 or 3/2 (d) – 2 or 3/2

[Answers : (1) d (2) a]


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 14

C O N C E P T S (Parabola)

C1 Conic Sections :

A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixedpoint (focus) is in a constant ratio (eccentricity e) to its perpendicular distance from a fixed straight line(directrix).

C2 General equation of a conic : Focal directrix property :

The general equation of a conic with focus (p, q) & directrix lx + my + n = 0 is :

(l2 + m2) [(x – p)2 + (y – q)2] = e2 (lx + my + n)2 ax2 + 2hxy + by2 + 2gx + 2fy + c = 0.

C3 Definition and Terminology

A parabola is the locus of a point, whose distance from a fixed point (focus) is equal to perpendiculardistance from a fixed straight line (directix). Four standard forms of the parabola are y2 = 4ax; y2 = –4ax;x2 = 4ay; x2 = –4ay

For parabola y2 = 4ax

(i) Vertex is (0, 0) (ii) focus is (a, 0)

(iii) Axis is y = 0 (iv) Directrix is x + a = 0

Focal Distance : The distance of a point on the parabola from the focus.

Focal Chord : A chord of the parabola, which passes through the focus. y2 = 4ax

Double Ordinate : A chord of the parabola perpendicular to the axis of the symmetry.

Latus Rectum : A double ordinate passing through the focus or a focal chord perpendicular to the axis ofparabola is called the Latus Rectum (L.R.)

Practice Problems :

1. The latus rectum of a parabola whose focal chord is P S Q such that S P = 3 and S Q = 2 is given by

(a)5

24(b)

5

12(c)

5

6(d) none of these

2. The length of the latus rectum of the parabola whose focus is (3, 3) and directrix is 3x – 4y – 2 = 0 is

(a) 2 (b) 1 (c) 4 (d) none of these

3. If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0, then one of the value of k is

(a) 1/8 (b) 8 (c) 4 (d) 1/4

4. y2 – 2x – 2y + 5 = 0 is

(a) a circle with centre (1, 1) (b) a parabola with vertex (1, 2)

(c) a parabola with directrix 2

3x (d) a parabola with directrix

2

1x

5. The vertex of the parabola x2 + 2y – 8x + 7 = 0 is

(a) (4, 11/2) (b) (4, 9/2) (c) (9/2, 4) (d) none of these

6. It (at2, 2at) are the co-ordinates of one end of a focal chord of the parabola y2 = 4ax, then theco-ordinates of the other end are

(a) (at2, – 2at) (b) (–at2, –2at) (c)

t

a2,

t

a2 (d)

t

a2,

t

a2

[Answers : (1) a (2) a (3) c (4) c (5) b (6) d]


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 15

C4 Line and a Parabola :

The line y = mx + c meets the parabola y2 = 4ax in two points real, coincident or imaginary according as a

cm condition of tangency is c = a/m.

Practice Problems :

1. The line which is parallel to x-axis and crosses the curve y = x at an angle of 450 is

(a) x = 1/4 (b) y = 1/4 (c) y = 1/2 (d) y = 1

[Answers : (1) c]

C5 Tangents to the Parabola y2 = 4ax :

(i) yy1 = 2a(x + x

1) at the point (x

1, y

1) ;

(ii)

m

a2,

m

aat)0m(

m

amxy

2

(iii) ty = x + at2 at (at2, 2at)

Important Point : Point of intersection of the tangents at the point t1 & t

2 is [at

1t2, a(t

1 + t

2)]

Practice Problems :

1. If y = m x + c touches the parabola y2 = 4a (x + a), then

(a)m

ac (b)

m

aamc (c)

m

aac (d) none of these

[Answers : (1) b]

C6 Normals to the parabola y2 = 4ax :

(i) )y,x(at)xx(a2

yyy 111

11 ;

(ii) y = mx – 2am – am3 at (am2 – 2am)

(iii) y + tx = 2at + at3 at (at2, 2at)

Practice Problems :

1. If x + y = k is normal to y2 = 12 x, then k is

(a) 3 (b) 9 (c) –9 (d) –3

2. Three normals to the parabola y2 = x are drawn through a point (c, 0) then

(a)4

1c (b)

2

1c (c)

2

1c (d) none of these

3. If the normal at (1, 2) on the parabola y2 = 4ax meets the parabola again at the point (t2, 2t), then thevalue of t is

(a) 1 (b) 3 (c) –3 (d) 1

4. If the normals from any point to the parabola y2 = 4x cuts the line x = 2 in points whose ordinate arein A.P., then the slopes of the tangents at 3 co-normal points are in

(a) A.P. (b) G.P. (c) H.P. (d) none of these

[Answers : (1) b (2) c (3) c (4) b]


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 16

C7 Pair of Tangents :

The equation to the pair of tangents which can be drawn from any point (x1, y

1) to the parabola y2 = 4ax is

given by : SS1 = T2 where :

S y2 – 4ax ; S1 = y

12 – 4ax

1; T y y

1 – 2a(x + x

1)

C8 Chord of Contact :

Equation to the chord of contact of tangents drawn from a point P(x1, y

1) is

yy1 = 2a (x + x

1).

Important Point : The area of the triangle formed by the tangents from the point (x1, y

1) & the chord of

contact is (y12 – 4ax

1)3/2 2a.

Practice Problems :

1. If the chord of contact of tangents drawn from a point P to the parabola y2 = 4ax touches theparabola x2 = 4by, then the locus of P is

(a) a circle (b) an ellipse (c) a parabola (d) a hyperbola

(d)

2. If a chord of the parabola y2 = 4ax which is normal at one end and which subtends a right angle atvertex then its slope is

(a)2

1(b) 2 (c) 2 (d) none of these

(b)

[Answers : (1) d (2) b]

C9 Chord with a given middle point :

Equation of the chord of the parabola y2 = 4ax whose middle point is

(x1, y

1) is y – y

1 = 11

1

ST)xx(y

a2


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 17

C O N C E P T S (Ellipse)

C1 Definitions

It is a locus of a point which moves in such a way that the ratio of its distance from a fixed point and a fixedline (not passes through fixed point and all points and line lies in same plane) is constant (e) which is lessthan one.

C2 Standard Equation

Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is 1b

y

a

x2

2

2

2

,

where a > b & b2 = a2(1 – e2).

Eccentricity : )1e0(,a

b1e

2

2

Focii : S (a e, 0) & S (– a e, 0).

Equations of Directrices : e

ax&

e

ax .

Major Axis : The line segment AA in which the focii S & S lie is of length 2a & is called the major axis

(a > b) of the ellipse.

Minor Axis : The y-axis intersects the ellipse in the points )b,0(B&)b,0(B . The line segment

BB is of length 2b(b < a) is called the minor axis of the ellipse.

Principal Axis : The major & minor axes together are called principal axis of the ellipse.

Vertices : Point of intersection of ellipse with major axis. A (–a, 0) & A (a, 0).

Focal Chord : A chord which passes through a focus is called a focal chord.

Double Ordinate : A chord perpendicular to the major axis is called a double ordinate.

Latus Rectum : The focal chord perpendicular to the major axis is called the latus rectum.

Centre : The point which bisects every chord of the conic drawn through it, is called the centre of the conic.

Practice Problems :

1. For the ellipse x2 + 4y2 = 9

(a) the ecentricity is 2

1(b) the latus rectum is

2

3

(c) a focus is (33, 0) (d) a directrix is x = 23

2. If S and S are two foci of an ellipse 1b

y

a

x2

2

2

2

(a < b) and P(x1, y

1) is a point on it, then SP + PS

is equal to

(a) 2a (b) 2b (c) a + e x1

(d) b + e y1

3. If the focal distance of an end of the minor axis of an ellipse (referred to its axes as the axes of x andy respectively) is 2k and the distance between its foci is 2h, then its equation is

(a) 1h

y

k

x2

2

2

2

(b) 1hk

y

k

x22

2

2

2

(c) 1kh

y

k

x22

2

2

2

(d) 1hk

y

k

x22

2

2

2


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 18

4. The eccentricity of the ellipse 1b

y

a

x2

2

2

2

, it being given that the length of its latusrectum is half of

its major axis is

(a) 1/2 (b) 1/2 (c) 2 (d) none of these

5. If P(x, y), F1 = (3, 0), F

2 = (–3, 0) and 16x2 + 25y2 = 400, then PF

1 + PF

2 equals

(a) 8 (b) 6 (c) 10 (d) 12

[Answers : (1) d (2) b (3) b (4) b (5) c]

C3 Position of a Point w.r.t. an Ellipse :

The point P(x1, y

1) lies outside, inside or on the ellipse according as ; 0or1

b

y

a

x2

21

2

21 .

C4 Line and an Ellipse :

The line y = mx + c meets the ellipse 1b

y

a

x2

2

2

2

in two points real, coincident or imaginary according

as c2 is < = or > a2m2 + b2.

Hence y = mx + c is tangent to the ellipse 1b

y

a

x2

2

2

2

= 1 if c2 = a2m2 + b2.

C5 Tangents :

(a) Slope form; 222 bmamxy is tangent to the ellipse 1

b

y

a

x2

2

2

2

for all values of

m.

(b) Point form : 1b

yy

a

xx21

21 is tangent to the ellipse 1

b

y

a

x2

2

2

2

at (x1, y

1)

(c) Parametric form : 1b

siny

a

cosx

is tangent to the ellipse 1

b

y

a

x2

2

2

2

at the point

(a cos , b sin ).

Practice Problems :

1. The number of values of c such that the straight line y = 4x + c touches the curve 1y4

x 22

is

(a) 0 (b) 1 (c) 2 (d) Infinite

2. A point on the ellipse 4x2 + 9y2 = 36 where the tangent is equally inclined to the axes is

(a)

13

4,

13

9(b)

13

4,

13

9

(c)

13

4,

13

9(d) all the above


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 19

3. On the ellipse 4x2 + 9y2 = 1, the points at which the tangents are parallel to the line 8x = 9y are

(a)

5

1,

5

2(b)

5

1,

5

2(c)

5

1,

5

2(d)

5

1,

5

2

[Answers : (1) c (2) d (3) b]

C6 Normals :

(i) Equation of the normal at (x1, y

1) to the ellipse 22

1

2

1

2

2

2

2

2

bay

yb

x

xais1

b

y

a

x .

(ii) Equation of the normal at the point (a cos . b sin ) to the ellipse 1b

y

a

x2

2

2

2

is;

ax . sec – by cosec = (a2 – b2).

(iii) Equation of a normal in terms of its slope ‘m’ is 222

22

mba

m)ba(mxy

.

Practice Problems :

1. If the normal at any point P on the ellipse 1b

y

a

x2

2

2

2

meets the axis in G and g respectively, then

P G : P g =

(a) a : b (b) a2 : b2 (c) b2 : a2 (d) b : a

2. The line lx + my + n = 0 is a normal to the ellipse 1b

y

a

x2

2

2

2

, if

(a) 2

222

2

2

2

2

n

)ba(

l

b

m

a (b) 2

222

2

2

2

2

n

)ba(

m

b

l

a

(c) 2

222

2

2

2

2

n

)ba(

m

b

l

a (d) none of the above

[Answers : (1) c (2) b]


The equation to the pair of tangents which can be drawn from any point (x1.y

1) to the ellipse 1

b

y

a

x2

2

2

2

is given by : SS1 = T2 where :

1b

y

a

xS

2

2

2

2

; 1b

y

a

xS

2

21

2

21

1 ; 1b

yy

a

xxT

21

21


Equation to the chord of contact of tangents drawn from a point P(x1.y

1) to the ellipse 1

b

y

a

x2

2

2

2

is T =

0, where 1b

yy

a

xxT

21

21 .


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 20


Equation of the chord of the ellipse 1b

y

a

x2

2

2

2

whose middle point is (x1, y

1) is T = S

1, where

1b

y

a

xS

2

21

2

21

1 ; 1b

yy

a

xxT

21

21

Practice Problems :

1. The locus of the middle point of the portion of the tangent of the ellipse 1b

y

a

x2

2

2

2

included

between the axes is the curve

(a) 4b

y

a

x2

2

2

2

(b) 4b

y

a

x2

2

2

2

(c) 4y

b

a

x2

2

2

2

(d) none of these

[Answers : (1) a]


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 21

C O N C E P T S (Hyperbola)

The Hyperbola is a conic whose eccentricity is greater than unity (e > 1)

C1 Standard Equation & Definition(s)

Standard equation of the hyperbola is 1b

y

a

x2

2

2

2

.

where b2 = a2(e2 – 1)

Eccentricity (e) :

2

22

a

b1e

Foci : S (ae, 0) & S (–ae, 0)

Practice Problems :

1. If e1 and e

2 are the ecentricities of the conic sections 16x2 + 9y2 = 144 and 9x2 – 16y2 = 144, then :

(a) e12 + e

22 = 3 (b) e

12 + e

22 > 3 (c) e

12 + e

22 < 3 (d) e

12 – e

22 = 1

[Answers : (1) c]

C2 Conjugate Hyperbola :

e.g. 1b

y

a

x&1

b

y

a

x2

2

2

2

2

2

2

2

are conjugate hyperbola of each.

Important Points :

(a) If e1 & e

2 are the eccentricities of the hyperbola & its conjugate then e

1–2 + e

2–2 = 1.

(b) The foci of a hyperbola and its conjugate are concyclic and form the vertices of a square.

(c) Two hyperbolas are said to be similar if they have the same eccentricity.

(d) Two similar hyperbolas are said to be equal if they have same latus rectum.

(e) If a hyperbola is equilateral then the conjugate hyperbola is also equilateral.

C3 Position of a Point ‘P’ w.r.t. A Hyperbola :

The quantity 1b

y

a

xS

2

21

2

21

1 is positive, zero or negative according as the point (x1, y

1) lies inside on

or outside the curve.

C4 Line And A Hyperbola :

The straight line y = mx + c is a secant, a tangent or passes outside the hyparbola 1b

y

a

x2

2

2

2

according

as : c2 > or = or < a2m2 – b2, respectively.

C5 Tangents :

(i) Slope Form : 222 bmamxy can be taken as the tangent to the hyperbola

1b

y

a

x2

2

2

2

, having slope ‘m’.


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 22

(ii) Point Form : Equation of tangent to the hyparbola 1b

y

a

x2

2

2

2

at the point (x1, y

1) is

1b

yy

a

xx21

21 .

(iii) Parametric Form : Equation of the tangent to the hyperbola 1b

y

a

x2

2

2

2

at the point.

(a sec , b tan ) is 1b

tany

a

secx

.

Practice Problems :

1. If the line y = 3x + touches the hyperbola 9x2 – 5y2 = 45, then the value of is

(a) 36 (b) 6 (c) 15 (d) 45

(b)

2. The line x cos + y sin = p touches the hyperbola 1b

y

a

x2

2

2

2

if

(a) a2 cos2 – b2sin2 = p2 (b) a2 cos2 – b2sin2 = p

(c) a2 cos2 + b2sin2 = p2 (d) a2 cos2 + b2sin2 = p

(d)

[Answers : (1) b (2) d]

C6. Normals :

(a) The equation of the normal to the hyperbola 1b

y

a

x2

2

2

2

at the point P(x1, y

1) on it is

2222

1

2

1

2

eabay

yb

x

xa

(b) The equation of the normal at the point P (a sec , b tan ) on the hyperbola 1b

y

a

x2

2

2

2

is

2222 eabatan

by

sec

ax

.

(c) Equation of normals in terms of the slope ‘m’ are 222

22

mba

m)ba(mxy

.


The equations to the pair of tangents which can be drawn from any point (x1, y

1) to the hyperbola

1b

y

a

x2

2

2

2

is given by SS1 = TT2 where :

1b

y

a

xS

2

2

2

2

; 1b

y

a

xS

2

21

2

21

1 ; 1b

yy

a

xxT

21

21


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 23

Practice Problems :1. If x = 9 is the chord of contact of the hyperbola x2 – y2 = 9, then the equation of the corresponding

pair of tangents is(a) 9x2 – 8y2 + 18x – 9 = 0 (b) 9x2 – 8y2 – 18x + 9 = 0(c) 9x2 – 8y2 – 18x – 9 = 0 (d) 9x2 – 8y2 + 18x + 9 = 0

2. Number of point (s) outside the hyperbola 136

y

25

x 22

from where two perpendicular tangents

can be drawn to the hyperbola is(a) zero (b) 1 (c) 2 (d) 3

3. The tangents from (1, 22) to the hyperbola 16x2 – 25y2 = 400 include between them an angle equalto

(a)6

(b)

3

(c)

2

(d)

4

[Answers : (1) b (2) a (3) c]


Equation to the chord of contact of tangents drawn from a point P(x1, y

1) to the hyperbola 1

b

y

a

x2

2

2

2

is

T = 0, where 1b

yy

a

xxT

21

21


Equation of the chord of the hyperbola 1b

y

a

x2

2

2

2

whose middle point is (x1, y

1) is T = S

1, where

1b

y

a

xS

2

21

2

21

1 ; 1b

yy

a

xxT

21

21 .

Practice Problems :1. Equation of the chord of the hyperbola 25x2 – 16y2 = 400 which is bisected at the point (6, 2) is

(a) 16x – 75y = 418 (b) 75x – 16y = 200(c) 25x – 4y = 400 (d) None of these

2. The combined equation of the asymptotes of the hyperbola 2x2 + 5xy + 2y2 + 4x + 5y = 0 is(a) 2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0 (b) 2x2 + 5xy + 2y2 + 4x + 5y – 2 = 0(c) 2x2 + 5xy + 2y2 = 0 (d) none of the above[Answers : (1) b (2) a]

C10 Rectangular or Equilateral Hyperbola :The particular kind of hyperbola in which the lengths of the transverse & conjugate axis are equal is calledan Equilateral Hyperbola.Practice Problems :

1. The equation of the chord joining two points (x1, y

1) and (x

2, y

2) on the rectangular hyperbola xy = c2

is

(a) 1yy

y

xx

x

2121

(b) 1

yy

y

xx

x

2121

(c) 1xx

y

yy

x

2121

(d) 1

xx

y

yy

x

2121

[Answers : (1) a]


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 24

INITIAL STEP EXERCISE

1. If the quadrilateral formed by the lines

ax + by + c = 0, 0cybxa have

perpendicular diagonals, then :

(a) b2 + c2 = 22 cb

(b) c2 + a2 = 22 ac

(c) a2 + b2 = 22 ba

(d) none of the above

2. If u = a1x + b

1y + c

1 = 0 and v = a

2x + b

2y + c

2 = 0 and

2

1

2

1

2

1

c

c

b

b

a

a , then u + kv = 0 represents :

(a) u = 0

(b) a family of concurrent lines

(c) a family of parallel lines


3. If 16a2 – 40ab + 25b2 – c2 = 0, then the lineax + by + c = 0 passes through the points :

(a) (4, –5) and (–4, 5)

(b) (5, –4) and (–5, 4)

(c) (1, –1) and (–1, 1)


4. If the abscissae and ordinates of two points P andQ are the roots of the equation x2 + 2ax – b2 = 0 andx2 + 2px – q2 = 0 respectively, the the equation ofthe circle with P Q as diameter is

(a) x2 + y2 + 2ax + 2py – b2 – q2 = 0

(b) x2 + y2 – 2ax – 2py + b2 + q2 = 0

(c) x2 + y2 – 2ax – 2py – b2 – q2 = 0

(d) x2 + y2 + 2ax + 2py + b2 + q2 = 0

5. The length of the common chord of the circles(x – a)2 + (y – b)2 = c2 and (x – b)2 + (y – a)2 = c2 is

(a) 22 )ba(c

(b) 22 )ba(2c4

(c) 22 )ba(c2

(d) 22 )ba(c4

6. If the circle x2 + y2 + 2gx + 2fy + c = 0 bisects thecircumference of the circle

0cyf2xg2yx 22 , then the length of the

common chord of these two circles is

(a) cfg2 22

(b) cfg2 22

(c) cfg2 22

(d) cfg2 22

7. The angle between the tangents from (a, b) to thecircle x2 + y2 = a2 is

(a)

1

1

S

atan

(b)

1

1

S

atan2

(c)

a

Stan2 11

(d) none of these

8. If

i

im

1,m , i = 1, 2, 3, 4 are concyclic points, then

the value of m1m

2m

3m

4 is

(a) 1 (b) –1

(c) 0 (d) none of these

9. If a line joining two points A(2, 0) and B(3, 1) isrotated about A in the anticlockwise directionthrough an angle 150, then the equation of the linein the new position is

(a) 3x – y = 23 (b) 3x + y = 23

(c) x + 3y = 23 (d) none of theabove

10. The triangle PQR is inscribed in the circlex2 + y2 = 25. If Q and R have co-ordinates (3, 4) and(–4, 3) respectively, then Q P R is equal to

(a) /2 (b) /3

(c) /4 (d) /6


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 25

11. If two distinct chords, drawn from the point (p, q)on the circle x2 + y2 – px – qy = 0 (where pq 0) arebisected by the x-axis, then

(a) p2 = q2 (b) p2 = 8q2

(c) p2 < 8q2 (d) p2 > 8q2

12. The centre of the circle passing through the points(0, 0), (1, 0) and touching the circle x2 + y2 = 9 is

(a)

2

1,

2

3(b)

2

3,

2

1

(c)

2

1,

2

1(d)

2/12,

2

1

13. If the circle x2 + y2 = a2 intersects the hyperbolaxy = c2 in four points P(x

1, y

1), Q(x

2, y

2), R(x

3, y

3)

and S(x4, y

4) then

(a) x1 + x

2 + x

3 + x

4 = 0

(b) y1 + y

2 + y

3 + y

4 = 0

(c) x1x

2x

3x

4 = c4

(d) All the above

14. If 2x – 4y = 9 and 6x – 12y + 7 = 0 are commontangents to a circle, then radius of the circle is

(a)5

3(b)

56

17

(c)3

2(d)

53

17

15. The number of integral values of m for which thex-co-ordinates of the point of intersection of the lines3x + 4y = 9 and y = mx + 1 is also an integer is

(a) 2 (b) 0

(c) 4 (d) 1

16. If PQ is a double ordinate of the hyperbola

1b

y

a

x2

2

2

2

such that OPQ is an equilateral

triangle, O being the centre of the hyperbola. Thenthe eccentricity e of the hyperbola satisfies

(a)3

2e1 (b)

3

2e

(c) 2/3e (d)3

2e

17. A (1, 3) and C(7, 5) are two opposite vertices of asquare. The equation of a side through A is

(a) x + 2y – 7 = 0 (b) x – 2y + 5 = 0

(c) 2x + y – 5 = 0 (d) 2x – y + 1 = 0

18. If P (1, 0), Q (–1, 0) and R (2, 0) are three givenpoints, then the locus of S satisfying the relationS Q2 + S R2 = 2S P2 is

(a) a straight line parallel to x-axis

(b) a circle through the origin

(c) a circle with centre at the origin

(d) a straight line parallel to y-axis

19. The lines x + (a – 1)y + 1 = 0 and 2x + a2y – 1 = 0 areperpendicular if

(a) |a| = 2 (b) 0 < a < 1

(c) –1 < a < 0 (d) a = –1

20. The lines y = mx, y + 2x = 0, y = 2x + k andy + mx = k form a rhombus if m equals

(a) –1 (b) 1/2

(c) 1 (d) 2

21. The equation of a straight line passing through thepoint (–5, 4) and which cuts off an intercept of2 units between the lines x + y + 1 = 0 andx + y – 1 = 0 is

(a) x – 2y + 13 = 0 (b) 2x – y + 14 = 0

(c) x – y + 9 = 0 (d) x – y + 10 = 0

22. The area bounded by the curves x + 2|y| = 1 andx = 0 is

(a) 1/4 (b) 1/2

(c) 1 (d) 2

23. The circle x2 + y2 = 4 cuts the circlex2 + y2 + 2x + 3y – 5 = 0 in A and B, then centre ofthe circle AB as diameter is

(a) (0, 0) (b)

13

3,

13

2

(c)

13

6,

13

4(d) (2, –1)

24. The equation of a circle passing from two points(1, 1) and (2, 2) having the radius 4 times theradius of smallest circle passing from above twopoints is :

(a) x2 + y2 + (15 – 3)x – (15 + 3)y + 4 = 0

(b) x2 + y2 – 3x – 3y + 4 = 0

(c) x2 + y2 – (15 + 3)x + (15 + 3)y + 4 = 0

(d) x2 + y2 – 6x – 6y + 4 = 0

25. A square is inscribed in the circle

x2 + y2 – 2x + 4y + 3 = 0

Its sides are parallel to the co-ordinates axes. Thenone vertex of the square is

(a) (1 + 2, – 2) (b) (1 – 2, – 2)

(c) (1, – 2 + 2) (d) None of these


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MCG – 26

26. The points (3, 2), (5, 3), (3 + 2t, 2 + t) and(3 + 2t2, 2 + t2) are

(a) vertices of a parallelogram

(b) vertices of a rectangle

(c) vertices of a square

(d) none of these

27. Let A = (3, –4), B = (1, 2) and P = (2k – 1, 2k + 1) isa variable point such that PA + PB is the minimum.Then k is

(a) 7/9 (b) 0

(c) 7/8 (d) none of these

28. The points

3

8,0 , (1, 3) and (82, 30) are vertices

of

(a) an obtuse angled triangle

(b) an acute angled triangle

(c) a right angled triangle

(d) none of these

29. All points inside the triangle formed by the points(1, 3), (5, 0) and (–1, 2) satisfy

(a) 3x + 2y 0

(b) 2x + y – 13 0

(c) 2x – 3y – 12 0

(d) both (a) and (c) are correct

30. Line L has intercepts a and b on the co-ordinateaxes. When the axes are rotated through a givenangle, keeping the origin fixed, the same line L hasintercepts p and q, then

(a) a2 + b2 = p2 + q2

(b) 2222 q

1

p

1

b

1

a

1

(c) a2 + p2 = b2 + q2

(d) 2222 q

1

b

1

p

1

a

1

31. The equation of the tangent to the ellipse4x2 + 3y2 = 5 which are parallel to the line y = 3x +7 are

(a)3

155x3y

(b)12

155x3y

(c)12

95x3y

(d) none of these

32. The equation of the base of an equilateral triangleis x + y = 2 and the vertex is (2, –1). Length of itsside is

(a)

2

1(b)

2

3

(c)

3

2(d) 2

33. Given four lines with equations x + 2y – 3 = 0,3x + 4y – 7 = 0, 2x + 3y – 4 = 0, 4x + 5y – 6 = 0, then

(a) they are all concurrent

(b) they are the sides of a quadrilateral

(c) both (a) and (b)

(d) None of these

34. A (–1, 1), B (5, 3) are opposite vertices of a squarein xy-plane. The equation of the other diagonal (notpassing through A, B) of the square is given by

(a) x – 3y + 4 = 0 (b) 2x – y + 3 = 0

(c) y + 3x – 8 = 0 (d) x + 2y – 1 = 0

35. The equation of the line which bisects the obtuseangle between the lines x – 2y + 4 = 0 and4x – 3y + 2 = 0 is

(a) (4 – 5)x – (3 – 25) y + (2 – 45) = 0

(b) (3 – 25)x – (4 – 5) y + (2 + 45) = 0

(c) (4 + 5)x – (3 + 25) y + (2 + 45) = 0

(d) None

36. From any point on the circle x2 + y2 = a2 tangentsare drawn to the circle x2 + y2 = a2 sin2. The anglebetween them is

(a) /2 (b)


37. The number of common tangents to the circlesx2 + y2 = 4 and x2 + y2 – 6x – 8y = 24 is

(a) 0 (b) 1

(c) 3 (d) 4

38. The locus of the mid-points of a chord of the circlex2 + y2 = 4 which subtends a right angle at theorigin is

(a) x + y = 2 (b) x2 + y2 = 1

(c) x2 + y2 = 2 (d) x + y = 1


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 27

39. The locus of the centre of a circle of radius 2 whichrolls on the outside of the circlex2 + y2 + 3x – 6y – 9 = 0 is

(a) x2 + y2 + 3x – 6y + 5 = 0

(b) x2 + y2 + 3x – 6y – 31 = 0

(c) x2 + y2 + 3x – 6y + 29/4 = 0

(d) none

40. The equation of the common tangent touching thecircle (x – 3)2 + y2 = 9 and the parabola y2 = 4x abovethe x-axis is

(a) 3 y = 3x + 1

(b) 3 y = –(x + 3)

(c) 3 y = x + 3

(d) 3 y = – (3x + 1)

41. The triangle formed by the tangent to the curvef(x) = x2 + bx – b at the point (1, 1) and the co-ordinates axes lies in the first quadrant. If its areais 2, then the value of b is

(a) –1 (b) 3

(c) –3 (d) 1

42. The tangents to the hyperbola x2 – y2 = 3 are paral-lel to the straight line 2x + y + 8 = 0 at the followingpoints

(a) (2, 1) (b) (2, –1)

(c) (–2, 1) (d) (–2, –1)

43. If the lines a1x + b

1y + c

1 = 0 and a

2x + b

2y + c

2 = 0

cut the co-ordinates axes is concylic points, then

(a) a1a

2 = b

1b

2(b) a

1b

1 = a

2b

2

(c) a1b

2 = a

2b

1(d) none of these

44. The ecentricity of the hyperbola whose latus rec-tum is 8 and conjugate axis is equal to half of thedistance between foci is

(a)3

2(b)

3

4

(c)3

4(d) none of these

45. The equation 1k8

y

k12

x 22

represents :

(a) a hyperbola if k < 8

(b) an ellipse if k > 8

(c) a hyperbola if 8 < k < 12


46. A common tangent to 9x2 – 16y2 = 144 andx2 + y2 = 9 is

(a)7

15x

7

3y

(b)7

15x

7

23y

(c) 715x7

32y


47. The co-ordinates of an end-point of the latus rec-tum of the parabola (y – 1)2 = 4(x + 1) are

(a) (0, –3) (b) (0, –1)

(c) (0, 1) (d) (1, 3)

48. An equilateral triangle is inscribed in the parabolay2 = 4ax whose one vertex is at the vertex of theparabola. Then length of its side is

(a) 4a 3 (b) 2a 3

(c) 16a 3 (d) 8a 3

49. If the segment intercepted by the parabola y2 = 4axwith the line lx + my + n = 0 subtends a right angleat the vertex, then

(a) 4al + n = 0

(b) 4al + 4am + n = 0

(c) 4am + n = 0

(d) al + n = 0

50. The ecentricity of the ellipse which meets the

straight line 12

y

7

x on the axis of x and the

straight line 15

y

3

x on the axis of y and whose

axes lie along the axes of co-ordinates is :

(a)7

23(b)

7

3

(c)7

62(d) none of these


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 28

51. If and are ecentric angles of the ends of a

focal chord of the ellipse 1b

y

a

x2

2

2

2

, then

2tan

2tan

is equal to

(a)e1

e1

(b)

1e

1e

(c)1e

1e

(d) none of these

52. The equation of the parabola whose vertex and thefocus lie on the axis of x at distances a and a

1 from

the origin respectively is

(a) y2 = 4(a1 – a) x

(b) y2 = 4(a1 – a) (x – a)

(c) y2 = 4 (a1 – a) (x – a

1)

(d) None of these

53. The locus of the point of intersection of normals atthe point on the parabola where tangents drawnmeet at the directrix is

(a) a parabola (b) a circle

(c) an ellipse (d) a hyperbola

54. The equation

6)3y(x)3y(x 2222 represents

(a) a circle (b) a line segment

(c) a pair of lines (d) none of these

55. x = sin2t, y = 2 sin t are the parametric equations of

(a) a parabola

(b) a portion of a parabola

(c) a hyperbola

(d) none of these

56. If two distinct chords of a parabola y2 = 4ax, pass-ing through (a, 2a) are bisected on the line x + y = 1,then length of the latus-rectum can be

(a) 6 (b) 1

(c) 4 (d) 5

57. The radius of the circle passing through the foci ofthe ellipse x2/16 + y2/9 = 1, and having its centre at(0, 3) is

(a) 4 (b) 3

(c) (1/2) (d) 7/2

58. Length of chord of parabola x2 = 4ay passingthrough vertex and having slope tan is

(a) 4|a| cosec cot

(b) 4|a| sec tan

(c) 4|a| cos cot

(d) 4|a| sin tan

59. If M is the foot of perpendicular drawn from a pointP on a parabola y2 = 4ax to its directrix and SPM isan equilateral triangle, where S is the focus, thenSP is equal to

(a) a (b) 2a

(c) 3a (d) 4a

60. The condition that the parabola y2 = 4c (x – d) andy2 = 4ax have a common normal other than x-axis(a > 0, c > 0) is

(a) 2a < 2c + d (b) 2a > 2c + d

(c) 2a < 2c – d (d) 2a > 2c – d

61. The equation 8x2 + 8xy + 2y2 + 26x + 13y + 15 = 0represents a pair of straight lines. The distancebetween them is

(a)5

7(b)

52

7

(c)5

7(d) none of these

62. The equation x2 + y2 + 4x + 6y + 13 = 0 represents

(a) a circle

(b) a pair of two straight lines

(c) a pair of coindent straight lines

(d) a point

63. The centre of the circle passing through points(0, 0), (1, 0) and touching the circle x2 + y2 = 9

(a) (3/2, 1/2) (b) (1/2, 3/2)

(c) (1/2, 1/2) (d) (1/2, –2½)

64. The curve described parametrically byx = t2 + t + 1, y = t2 – t + 1 represents :

(a) a pair of straight lines

(b) an ellipse

(c) a parabola

(d) a hyperbola

65. The parametric representation (2 + t2, 2t + 1)represents

(a) a parabola with focus S at (2, 1)

(b) a parabola with vertex at (2, 1)

(c) an ellipse with centre at (2, 1)



New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 29

FINAL STEP EXERCISE

1. If a variable line drawn through the point of

intersection of straight lines 1yx

and

1yx

meets the co-ordinates axes in A and B,

then the locus of the mid point of A B is

(a) (x + y) = x y ( + )

(b) (x + y) = 2 x y ( + )

(c) ( + ) (x + y) = 2 x y


2. The area of the parallelogram formed by the lines3x – 4y + 1 = 0, 3x – 4y + 3 = 0, 4x – 3y – 1 = 0 and4x – 3y – 2 = 0 is

(a)6

1 square units

b)7

2 square units

(c)8

3 square units

(d) none of these

3. If a variable line passes through the point ofintersection of the lines x + 2y – 1 = 0 and2x – y – 1 = 0 and meets the co-ordinates axes in Aand B, then the locus of the mid point of AB is

(a) x + 3y = 0 (b) x + 3y = 10

(c) x + 3y = 10xy (d) none of these

4. If a chord of a circle x2 + y2 = 8 makes equalintercepts of length ‘a’ on the co-ordinates axes, then

(a) |a| < 8 (b) |a| < 42

(c) |a| < 4 (d) |a| > 4

5. The equation of the tangent to the circle x2 + y2 = a2

which makes with the axes a triangle of area a2 is

(a) x ± y = 2a (b) x ± y = 2 a

(c) x + y = 2a2 (d) none of these

6. If a circle of constant radius 3k passes through theorigin and meets the axes at A and B, the locus ofthe centroid of OAB is

(a) x2 + y2 = k2 (b) x2 + y2 = 2k2

(c) x2 + y2 = 3k2 (d) none of theabove

7. The tangents to x2 + y2 = a2 having inclination and intersect at P. If cot + cot = 0, then thelocus of P is

(a) x + y = 0 (b) x – y = 0

(c) x y = 0 (d) none of these

8. The distance between the chords of the tangents tothe circle x2 + y2 + 2gx + 2fy + c = 0 from the originand the point (g, f) is

(a) g2 + f2 (b) )cfg(2

1 22

(c)22

22

fg

cfg.

2

1

(d)

22

22

fg

cfg.

2

1

9. The circle x2 + y2 – 6x – 10y + c = 0 does notintersect or touch both axes of co-ordinates and thepoint (1, 4) lies inside the circle. Then range ofpossible values of c is given by

(a) 25 < c < 29 (b) c > 29

(c) c > 25 (d) c < 25

10. If the circle C1 : x2 + y2 = 16 intersects another circle

C2 of radius 5 in such a manner that the common

chord is of maximum length and has a slope equalto 3/4, the co-ordinates of the centre of C

2 are

(a)

5

12,

5

9;

5

12,

5

9

(b)

5

12,

5

9;

5

12,

5

9

(c)

5

9,

5

12;

5

9,

5

12


11. If the straight lines ax + by + c = 0 and x cos + ysin = c enclose an angle /4 between them andmeet the straight line x sin – y cos = 0 in thesame point, then

(a) a2 + b2 = c2 (b) a2 + b2 = 2

(c) a2 + b2 = 2c2 (d) a2 + b2 = 4

12. The point A(2, 1) is translated parallel to the linex – y = 3 by a distance 4 units. If the new position

A is in the third quadrant, then co-ordinates of

A aree

(a) (2 + 22, 1 + 22)

(b) (– 2 + 2, – 1 – 22)

(c) (2 – 22, 1 – 22)

(d) both (a) and (c) are correct


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 30

13. The point of intersection of the common chords ofthree circles described on the three sides of atriangle as diameter is

(a) centroid of the triangle

(b) orthocentre of the triangle

(c) circumcentre of the triangle

(d) incentre of the triangle

14. Let PQ and RS be tangents at the extremities of thediameter PR of a circle of radius r. If PS and RQintersect at a point X on the circumference of thecircle, then 2r equals

(a) RS.PQ (b)2

RSPQ

(c)RSPQ

RS.PQ2

(d)

2

RSPQ 22

15. Let AB be a chord of the circle x2 + y2 = r2

subtending a right angle at the centre. Then thelocus of the centroid of the triangle PAB as P moveson the circle is

(a) a parabola (b) a circle

(c) an ellipse (d) a pair ofstraight lines

16. Let L be a straight line passing through the originand L

2 be the straight line x + y = 1. If the

intercepts made by the circle x2 + y2 – x + 3y = 0 onL

1 and L

2 are equal, then which of the following

equations can represent L1 ?

(a) x + y = 0 (b) x – y = 0

(c) x + 7y = 0 (d) x – 7y = 0

17. Let P Q R be a right angled triangle isosceles rightangled at P (2, 1). If the equation of the line Q R is2x + y = 3, then the equation of the pair of linesP Q and P R is

(a) 3x2 – 3y2 + 8xy + 20x + 10y + 25 = 0

(b) 3x2 – 3y2 + 8xy – 20x – 10y + 25 = 0

(c) 3x2 – 3y2 + 8xy + 10x + 15y + 20 = 0

(d) 3x2 – 3y2 – 8xy – 10x – 15y – 20 = 0

18. The equation to a pair of opposite sides of aparallelogram are x2 – 5x + 6 = 0 and y2 – 6y + 5 = 0.The equations to its diagonals are

(a) x + y = 13 and y = 4x – 7

(b) 4x + y = 13 and 4y = x – 7

(c) 4x + y = 13 and y = 4x – 7

(d) y – 4x = 13 and y + 4x = 7

19. If one of the diagonals of a square is along the linex = 2y and one of the vertices is (3, 0), then its sidesthrough this vertex are given by the equations

(a) y – 3x + 9 = 0, 3y + x – 3 = 0

(b) y + 3x + 9 = 0, 3y + x – 3 = 0

(c) y – 3x + 9 = 0, 3y – x + 3 = 0

(d) y – 3x + 3 = 0, 3y + x + 9 = 0

20. Area of the parallelogram formed by the linesy = mx, y = mx + 1, y = nx and y = nx + 1 equals

(a) 2)nm(

|nm|

(b)

|nm|

2

(c)|nm|

1

(d)

|nm|

1

21. AB is a diameter of a circle and C is any point onthe circumference of the circle. Then

(a) The area of ABC is maximum when itis isosceles

(b) The area of ABC is minimum when it isisosceles

(c) The perimeter of ABC is maximumwhen it is isosceles

(d) None of these

22. The image of the pair of lines represented byax2 + 2hxy + by2 = 0 by the line mirror y = 0 is

(a) ax2 – 2hxy – by2 = 0

(b) bx2 – 2hxy + ay2 = 0

(c) bx2 + 2hxy + ay2 = 0

(d) ax2 – 2hxy + by2 = 0

23. The point (4, 1) undergoes the following threetransformations sucessively

(a) Reflection about the line y = x

(b) Transformation through a distance 2units along the positive direction of x-axis

(c) Rotation through angle /4 about theorigin in the anticlockwise direction. Thefinal position of the point is given by thecoordinates

(a)

2

7,

2

1(b) (–2, 72)

(c)

2

7,

2

1(d) (2, 72)


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MCG – 31

24. If p1, p

2, p

3 be the perpendiculars from the point

(m2, 2m), (mm’, m + m’) and (m’2, 2m’) respectively

on the line 0cos

sinsinycosx

2

then

p1, p

2, p

3 are in

(a) A.P. (b) G.P.

(c) H.P. (d) None

25. Consider a circle with its centre lying on the focusof the parabola y2 = 2p x such that it touches thedirectrix of the parabola. Then, a point of intersec-tion of the circle and the parabola is

(a)

p,

2

p(b)

p,

2

p

(c)

p,

2

p(d)

p,

2

p

26. Let E be the ellipse 14

y

9

x 22

and C be the circle

x2 + y2 = 9. Let P and Q be the points (1, 2) and (2, 1)respectively. Then :

(a) Q lies inside C but outside E

(b) Q lies outside both C and E

(c) P lies inside both C and E

(d) P lies inside C but outside E

27. An ellipse has ecentricity 2

1 and one focus at the

point

1,

2

1P . Its one directrix is the common tan-

gent, nearer to the point P to the circle x2 + y2 = 1and the hyperbola x2 – y2 = 1. The equation of theellipse in standard form is

(a) 1)1y(123

1x9 2

2

(b) 1)1y(123

1x9 2

2

(c) 1)1y(123

1x9 2

2

(d) 1)1y(123

1x9 2

2

28. If C is the centre and A and B are the points on theconic 4x2 + 9y2 – 8x – 36 y + 4 = 0 such that

2BCA

, then CA–2 + CB–2 is equal to

(a)36

5(b)

36

13

(c)36

15(d)

36

16

29. If the normal at

t

c,ct on the curve xy = c2 meets

the curve again in 't' then

(a) 3t

1t (b)

t

1t

(c) 2t

1t (d) 2

2

t

1t

30. The two parabolas y2 = 4x and x2 = 4y intersect at apoint P, whose abscissa is not zero, such that

(a) they both touches each other at P

(b) they cut at right angles at P

(c) the tangents to each other at P makecomplementary angles with the x-axis


31. If P S Q is the focal chord of the parabola y2 = 8 xsuch that S P = 6. Then length S Q is

(a) 6 (b) 4(c) 3 (d) none of these

32. The point on the curve y2 = ax, the tangent at whichmakes an angle of 450 with x-axis will be given by

(a)

4

a,

2

a(b)

4

a,

2

a

(c)

2

a,

4

a(d)

2

a,

4

a

33. P is a variable point on the ellipse 1b

y

a

x2

2

2

2

with

AA as the major axis, then the maximum value of

the area of the triangle AAP is

(a) ab (b) 2ab(c) ab/2 (d) none of these


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MCG – 32

34. The line x = at2, meets the ellipse 1b

y

a

x2

2

2

2

in

real points iff

(a) |t| < 2 (b) |t| > 1

(c) |t| 1 (d) none of these

35. If chords of contact of tangents from two points

(x1, y

1) and (x

2, y

2) to the ellipse 1

b

y

a

x2

2

2

2

are at

right angles, then 21

21

yy

xx is equal to

(a) 2

2

b

a(b) 2

2

a

b

(c) 4

4

b

a (d) 4

4

a

b

36. If 1 and

2 are the ordinates of two points A and B

on the parabola and 3 is the ordinate of the point

of intersection of tangents at A and B, then

(a) 1,

2,

3 are in A.P.

(b) 1,

2,

3 are in G.P.

(c) 1,

2,

3 are in H.P.

(d) none of these

37. The centre of a circle passing through the point(0, 1) and touching the curve y = x2 at (2, 4) is

(a)

10

27,

5

16(b)

10

5,

7

16

(c)

10

53,

5

16(d) None of these

38. From any point on the hyperbola 1b

y

a

x2

2

2

2

,

tangents are drawn to the hyperbola 2b

y

a

x2

2

2

2

.

The area cut off by the chord of contact on the as-ymptotes is equal to

(a)2

ab(b) ab

(c) 2ab (d) 4ab

39. If a straight line L is perpendicular to the line5x – y = 1 such that area of the formed by the lineL and the co-ordinates axes is 5, then the equationof the line L is

(a) x + 5y + 5 = 0 (b) x + 5y ± 2 = 0

(c) x + 5y ± 5 = 0 (d) x + 5y ±2 = 0

40. If , and are the parametric angles of threepoints P, Q and R respectively, on the circlex2 + y2 = 1 and A is the point (–1, 0). If the length ofthe chords AP, AQ, AR are in G.P., then

2cos,

2cos,

2cos

are in

(a) A.P. (b) G.P.

(c) H.P. (d) none of these


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MCG – 33

ANSWERS (FINAL STEP EXERCISE)

1. b

2. b

3. c

4. c

5. b

6. d

7. c

8. d

9. a

10. a

11. b

12. d

13. b

14. a

15. b

16. b

17. b

18. c

19. a

20. d

21. a

22. d

23. c

24. b

25. a

26. d

27. b

28. a

29. a

30. b

31. c

32. c

33. a

34. c

35. a

36. d

37. c

38. d

39. d

40. b

ANSWERS (INITIAL STEP EXERCISE)

1. c2. a3. a4. a5. b6. a7. b8. a9. a10. c11. d12. d13. d14. b15. a16. d17. c18. d19. d20. d

21. c22. b23. b24. a25. d26. d27. c28. d29. d30. b31. b32. c33. c34. c35. a36. c37. b38. c39. b40. c

41. c42. b43. a44. a45. c46. b47. b48. d49. a50. c51. b52. b53. a54. b55. b56. b57. a58. b59. d60. a

61. b62. d63. d64. c65. a


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MCG – 34

AIEEE ANALYSIS [2002]

1. The greatest distance of the point P (10, 7) from thecircle

x2 + y2 – 4x – 2y – 20 = 0 is

(a) 10 (b) 5


2. The incentre of the triangle with vertices (1, 3),(0, 0) and (2, 0) is

(a)

3

1,1 (b)

2

3,

2

1

(c)

2

3,1 (d)

3

1,

3

2

3. Three straight lines

2x + 11y – 5 = 0, 24x + 7y – 20 = 0 and4x – 3y – 2 = 0

(a) form a triangle

(b) are only concurrent

(c) are concurrent with one line bisecting theangle between the other two

(d) none of these

4. The equation to the ellipse whose foci are (± 2, 0)

and eccentricity is 2

1 is

(a) 116

y

12

x 22

(b) 18

y

16

x 22

(c) 112

y

16

x 22

(d) none of these

5. A straight line through the point (2, 2) intersectsthe lines 3x + y = 0 and 3x – y = 0 at the points Aand B. The equation to the line AB so that the OABis equilateral is

(a) y – 2 = 0 (b) x – 2 = 0

(c) x + y – 4 = 0 (d) none of these

6. The radius of the circle passing through the foci of

the ellipse 19

y

16

x 22

and having its centre at

(0, 3) is

(a) 4 (b) 7/2

(c) 3 (d) 12

7. The equation of the chord joining two points(x

1, y

1) and (x

2, y

2) on the rectangular hyperbola

xy = c2 is

(a) 1yy

y

xx

x

2121

(b) 1yy

y

xx

x

2121

(c) 1xx

y

yy

x

2121

(d) 1xx

y

yy

x

2121

8. A triangle with vertices (4, 0), (–1, –1), (3, 5) is

(a) isosceles and right angled

(b) right angled but not isosceles

(c) neither right angled nor isosceles

(d) isosceles but not right angled

9. The equation of the directrix of the parabolay2 + 4y + 4x + 2 = 0 is

(a) x = –1 (b) x = 3/2

(c) x = –3/2 (d) x = 1

10. The equation of the tangent to the circlex2 + y2 + 4x – 4y + 4 = 0 which make equalintercepts on the positive coordinates axes is

(a) x + y = 2 (b) x + y = 22

(c) x + y = 4 (d) x + y = 8


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MCG – 35


11. The foci of the ellipse 1b

y

16

x2

22

and the

hyperbola 25

1

81

y

144

x 22

coincide. Then the

value of b2 is

(a) 5 (b) 7

(c) 9 (d) 1

12. The normal at the point (bt12, 2bt

1) on a parabola

meets the parabola again in the point (bt22, 2bt

2),

then

(a)1

12t

2tt

(b)1

12t

2tt

(c)1

12t

2tt

(d)1

12t

2tt

13. If the two circles (x – 1)2 + (y – 3)2 = r2 andx2 + y2 – 8x + 2y + 8 = 0 intersect in two distinctpoints, then

(a) r < 2 (b) r = 2

(c) r > 2 (d) 2 < r < 8

14. The lines 2x – 3y = 5 and 3x – 4y = 7 are diametersof a circle having area as 154 sq. units. Then theequation of the circle is

(a) x2 + y2 + 2x – 2y = 47

(b) x2 + y2 – 2x + 2y = 47

(c) x2 + y2 – 2x + 2y = 62

(d) x2 + y2 + 2x – 2y = 62

15. A square of side a lies above the x-axis and has onevertex at the origin. The side passing through the

origin makes an angle

40 with the

positive direction of x-axis. The equation of itsdiagonal not passing through the origin is

(a) y(cos + sin ) + x(sin – cos ) = a

(b) y(cos + sin ) + x(sin + cos ) = a

(c) y(cos + sin ) + x(cos – sin ) = a

(d) y(cos – sin ) – x(sin – cos ) = a

16. If the pair of straight lines x2 – 2pxy – y2 = 0 andx2 – 2qxy – y2 = 0 be such that each pair bisects theangle between the other pair, then

(a) p = –q (b) pq = 1

(c) pq = –1 (d) p = q

17. Locus of centroid of the triangle whose vertices are(a cos t, a sin t), (b sin t, – b cos t) and (1, 0), wheret is a parameter, is

(a) (3x – 1)2 + (3y)2 = a2 + b2

(b) (3x + 1)2 + (3y)2 = a2 + b2

(c) (3x + 1)2 + (3y)2 = a2 – b2

(d) (3x – 1)2 + (3y)2 = a2 – b2

18. If x1, x

2, x

3 and y

1, y

2, y

3 are both in G.P. with the

same common ratio, then the points

(x1, y

1), (x

2, y

2) and (x

3, y

3)

(a) lie on an ellipse

(b) lie on circle

(c) are vertices of triangle

(d) lie on a straight line

19. If the equation the locus of point equidistant fromthe point (a

1, b

1) and (a

2, b

2) is (a

1 –a

2)x + (b

1 – b

2) y

+ c = 0, then the value of ‘c’ is

(a) 22

21

22

21 bbaa

(b) )bbaa(2

1 22

21

22

21

(c) 22

22

21

21 baba

(d) 21

21

22

22 baba

2

1


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 36

AIEEE ANALYSIS [2004/2005]

20. Let A (2, –3) and B(–2, 1) be vertices of a triangleABC. If the centroid of this triangle moves on theline 2x + 3y = 1, then the locus of the vertex C is theline(a) 3x + 2y = 5 (b) 2x – 3y = 7(c) 2x + 3y = 9 (d) 3x – 2y = 3[2004]

21. The equation of the straight line passing throughthe point (4, 3) and making intercepts on theco-ordinates axes whose sum is –1 is

(a) 11

y

2

xand1

3

y

2

x

(b) 11

y

2

xand1

3

y

2

x

(c) 11

y

2

xand1

3

y

2

x

(d) 11

y

2

xand1

3

y

2

x

[2004]22. If the sum of the slopes of the line given by

x2 – 2cxy – 7y2 = 0 is four times their product, thenc has the value(a) 2 (b) –1(c) 1 (d) –2[2004]

23. If one of the lines given by 6x2 – xy + 4cy2 = 0 is3x + 4y = 0, then c equals(a) 3 (b) –1(c) 1 (d) –3[2004]

24. If a circle passes through the point (a, b) and cutsthe circle x2 + y2 = 4 orthogonally, then the locus ofits centre is(a) 2ax – 2by + (a2 + b2 + 4) = 0(b) 2ax + 2by – (a2 + b2 + 4) = 0(c) 2ax + 2by + (a2 + b2 + 4) = 0(d) 2ax – 2by – (a2 + b2 + 4) = 0[2004]

25. A variable circle passes through the fixed pointA(p, q) and touches x-axis. The locus of the otherend of the diameter through A is(a) (y – p)2 = 4qx (b) (x – q)2 = 4py(c) (x – p)2 = 4qy (d) (y – q)2 = 4px[2004]

26. If the lines 2x + 3y + 1 = 0 and 3x – y – 4 = 0 lie alongdiameters of a circle of circumference 10, then theequation of the circle is(a) x2 + y2 + 2x + 2y – 23 = 0(b) x2 + y2 – 2x – 2y – 23 = 0(c) x2 + y2 – 2x + 2y – 23 = 0(d) x2 + y2 + 2x – 2y – 23 = 0[2004]

27. The intercept on the line y = x by the circlex2 + y2 – 2x = 0 is AB. Equation of the circle on ABas a diameter is(a) x2 + y2 + x + y = 0(b) x2 + y2 – x + y = 0(c) x2 + y2 – x – y = 0(d) x2 + y2 + x – y = 0[2004]

28. The line parallel to the x-axis and passing throughthe intersection of the lines ax + 2by + 3b = 0 andbx – 2ay – 3a = 0 where (a, b) (0, 0) is

(a) below the x-axis at a distance of 3

2 from

it

(b) below the x-axis at a distance of 2

3 from

it

(c) above the x-axis at a distance of 3

2 from

it

(d) above the x-axis at a distance of 2

3 from

it[2005]

29. If non-zero numbers a, b, c are in H.P., then the

straight line 0c

1

b

y

a

x always passes through

a fixed point, that point is(a) (–1, –2) (b) (–1, 2)

(c) (1, 2

1 ) (d) (1, –2)

[2005]30. If a vertex of a triangle is (1, 1) and the mid points

of two sides through this vertex are (–1, 2) and(3, 2), then the centroid of the triangle is

(a)

3

7,

3

1(b)

3

7,1

(c)

3

7,

3

1(d)

3

7,1

[2005]31. If the circles x2 + y2 + 2ax + cy + a = 0 and

x2 + y2 – 3ax + dy – 1 = 0 intersect in two distinctpoints P and Q then the line 5x + by – a = 0 passesthrough P and Q for(a) no value of a(b) exactly one value of a


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MCG – 37


41. The locus of the vertices of the family of parabolas

a22

xa

3

xay

223

is

(a)105

64xy (b)

64

105xy

(c)4

3xy (d)

16

35xy

42. A straight line through the point A(3, 4) is such thatits intercept between the axes is bisected at A. Itsequation is

(a) 3x + 4y = 25 (b) x + y = 7

(c) 3x – 4y + 7 = 0 (d) 4x + 3y = 24

43. In an ellipse, the distance between its foci is 6 andminor axis is 8. Then its eccentricity is

(a) 1/5 (b) 3/5

(c) 1/2 (d) 4/5

(c) exactly two values of a(d) infinitely many values of a[2005]

32. A circle touches the x-axis and also touches the circlewith centre at (0, 3) and radius 2, the locus of thecentre of the circle is(a) a circle (b) an ellipse(c) a parabola (d) a hyperbola[2005]

33. If a circle passes through the point (a, b) and cutsthe circle x2 + y2 = p2 orthogonally, then theequation of the locus of its centre is(a) 2ax + 2by – (a2 – b2 + p2) = 0(b) x2 + y2 – 3ax – 4by + (a2 + b2 – p2) = 0(c) 2ax + 2by – (a2 + b2 + p2) = 0(d) x2 + y2 – 2ax – 3by + (a2 – b2 – p2) = 0[2005]

34. If the pair of lines ax2 + 2(a + b)xy + by2 = 0 liealong diameters of a circle and divide the circle intofour sectors such that the area of one of the sectorsin thrice the area of another sector then(a) 3a2 – 2ab + 3b2 = 0(b) 3a2 – 10ab + 3b2 = 0(c) 3a2 + 2ab + 3b2 = 0(d) 3a2 + 10ab + 3b2 = 0[2005]

35. If a 0 and the line 2bx + 3cy + 4d = 0 passesthrough the point of intersection of the parabolay2 = 4ax and x2 = 4ay, then(a) d2 + (2b – 3c)2 = 0(b) d2 + (3b + 2c)2 = 0(c) d2(2b + 3c)2 = 0(d) d2 + (2b + 3c)2 = 0[2004]

36. The eccentricity of an ellipse, with its centre at theorigin, is 1/2. If one of the directrices is x = 4, thenthe equation of the ellipse is(a) 4x2 + 3y2 = 12 (b) 3x2 + 4y2 = 12(c) 3x2 + 4y2 = 1 (d) 4x2 + 3y2 = 1[2004]

37. Area of the greatest rectangle that can be inscribed

in the ellipse 1b

y

a

x2

2

2

2

is

(a) ab (b) 2ab

(c)b

a(d) ab

[2005]38. Let P be the point (1, 0) and Q a point on y2 = 8x.

The locus of mid point of PQ is(a) y2 + 4x + 2 = 0 (b) y2 – 4x + 2 = 0(c) x2 – 4y + 2 = 0 (d) x2 + 4y + 2 = 0[2005]

39. An ellipse has OB as semi minor axis, F and F its

foci and the angle FFB is a right angle. Then theeccentricity of the ellipse is

(a)2

1(b)

2

1

(c)3

1(d)

4

1

[2005]40. The locus of a point P (, ) moving under the con-

dition that the line y = x + is a tangent to the

hyperbola 1b

y

a

x2

2

2

2

is

(a) a circle (b) an ellipse(c) a hyperbola (d) a parabola[2005]


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MCG – 38


47. For the hyperbola 1sin

y

cos

x2

2

2

2

, which of the

following remains constant when varies ?(a) Abscissae of vertices(b) Abscissae of foci(c) Eccentricity(d) Directrix

48. The equation of a tangent to the parabola y2 = 8x isy = x + 2. The point on this line from which theother tangent to the parabola is perpendicular tothe given tangent is(a) (2, 4) (b) (–2, 0)(c) (–1, 1) (d) (0, 2)

49. Let A (h, k), B (1, 1) and C (2, 1) be the vertices of aright angled triangle with AC as its hypotenuse. Ifthe area of the triangle is 1, then the set of valueswhich ‘k’ can take is given by(a) {–1, 3} (b) {–3, –2}(c) {1, 3} (d) (0, 2}

50. Let P = (–1, 0) Q = (0, 0) and R = (3, 33) be threepoints. The equation of the bisector of the anglePQR is

(a) 0yx2

3 (b) x + 3 y = 0

(c) 3 x + y = 0 (d) 0y2

3x

51. If one of the lines of my2 + (1 – m2) xy – mx2 = 0 is abisector of the angle between the lines xy = 0, thenm is(a) ± 1 (b) 2(c) –1/2 (d) –2

52. Consider a family of circles which are passingthrough the point (–1, 1) and are tangent to x-axis.If (h, k) are the co-ordinates of the centre of thecircles, then the set of values of k is given by theinterval(a) –½ k ½ (b) k ½(c) 0 < k < ½ (d) k ½

ANSWERS AIEEE ANALYSIS

1. c 2. a 3. c 4. c 5. a 6. a 7. a8. a 9. b 10. b 11. b 12. a 13. d 14. b15. c 16. c 17. a 18. d 19. d 20. c 21. d22. a 23. d 24. b 25. c 26. c 27. c 28. b29. d 30. d 31. a 32. c 33. c 34. c 35. c36. b 37. b 38. b 39. b 40. c 41. b 42. d43. b 44. a 45. a 46. d 47. b 48. b 49. a50. c 51. a 52. d

44. Let C be the circle with centre (0, 0) and radius 3units. The equation of the locus of the mid point ofthe chords of the circle C that subtend an angle of

3

2 at its centre is

(a) x2 + y2 = 9/4 (b) x2 + y2 = 3/2

(c) x2 + y2 = 1 (d) x2 + y2 = 27/4

45. If the lines 3x – 4y – 7 = 0 and 2x – 3y – 5 = 0 are twodiameters of a circle of area 49 square units, theequation of the circle is

(a) x2 + y2 – 2x + 2y – 47 = 0

(b) x2 + y2 + 2x – 2y – 47 = 0

(c) x2 + y2 + 2x – 2y – 62 = 0

(d) x2 + y2 – 2x + 2y – 62 = 0

46. If (a, a2) falls inside the angle made by the linear

equation 2

xy , x > 0 and y = 3x, x > 0, then a

belongs to

(a)

2

1,3 (b)

2

1,0

(c) (3, ) (d)

3,

2

1


New Delhi – 110 018, Ph. : 9312629035, 8527112111

MCG – 39

TEST YOURSELF

1. The straight lines x + y – 4 = 0, 3x + y – 4 = 0 andx + 3y – 4 = 0 form a triangle which is

(a) isosceles (b) right angled

(c) equilateral (d) none of these

2. The centroid of a triangle lies at the origin and thecoordinates of its two vertices are (–8, 7) and (9, 4).The area of the triangle is

(a) 95/6 (b) 285/2

(c) 190/3 (d) 285

3. Two equal sides of an isosceles triangle are givenby 7x – y + 3 = 0 and x + y – 3 = 0 and the third sidepasses through the point (1, 10), the slope m of thethird side is given by

(a) m2 – 1 = 0

(b) m2 – 3 = 0

(c) 3m2 – 1 = 0

(d) 3m2 + 8m – 3 = 0

4. The equation ax2 + 2hxy + ay2 = 0 represents a pairof coincident lines through the origin if

(a) h = 2a (b) 2h = a

(c) h2 = a (d) h2 = a2

5. An equation of the normal at the point (2, 3) to thecircle x2 + y2 – 2x – 2y – 3 = 0 is

(a) 2x + y – 7 = 0 (b) x + 2y – 3 = 0

(c) 2x – y – 1 = 0 (d) x – 2y + 1 = 0

6. The number of circles touching the coordinates axesand the line x + y = 1 is

(a) exactly one (b) two

(c) three (d) four

7. If (a cos i, a sin

i) i = 1, 2, 3 represent the vertices

of an equilateral triangle inscribed in a circle, then

(a) cos 1 + cos

2 + cos

3 = 0

(b) sin 1 + sin

2 + sin

3 0

(c) tan 1 + tan

2 + tan

3 = 0

(d) cot 1 + cot

2 + cot

3 = 0

8. The length of the chord of the parabola y2 = 4axwhose equation is y – x2 + 4a2 = 0 is

(a) a112 (b) a24

(c) a28 (d) a36

ANSWERS

1. a

2. b

3. d

4. d

5. c

6. c

7. a

8. d

9. c

10. d

9. If the area of the triangle inscribed in the parabolay2 = 4ax with one vertex at the vertex of theparabola and other two vertices at the extremitiesof a focal chord is 5a2/2, then the length of the focalchord is

(a) 3a (b) 5a

(c) 25a/4 (d) none of these

10. The coordinates of a point common to a directrixand an asymptote of the hyperbola x2/25 – y2/16 = 1are

(a) )3/20,41/25(

(b) )41/25,41/20(

(c) (25/3, 20/3)

(d) )41/20,41/25(

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CO-ORDINATE GEOMETRYeinsteinclasses.com/JEE Main Website/MATHEMATICS/Co...its equation, translation of axes, slope of a line, parallel and perpendicular lines, intercepts of a line