Tutorials '9892603993 MH-CET/XII Mathematics

5 C O N I C SA conic is the locus of a point such that the ratio of its distance from a fixed point to its distance

from a fixed straight line is constant.

The fixed point is called focus of the conic and is denoted by S. the fixed straight line is called

directrix of the conics and is denoted by d . The constant ratio is called the eccentricity of the

conic and is denoted by e

If S is the focus , d is the directrix P ( x1, y1) is any point on the conic and PM d , the equation

of the conic is given by PS = e PM .

This property of the conics is called focus-directrix property .

If e = 1 , the conic is called a Parabola

If e < 1 , the conic is called a Ellipse

If e > 1 , the conic is called a Hyperbola

The general equation of a conic is given by a

general second degree equation in x and y.

Important Term in Conics :

Axis : A line about which a conic is symmetrical is called an axis of the conic

Vertex : The point in which an axis of a conic meets the curve is called a vertex of the conics.

Focal point : The distance of a point on a conic from its focus is called focal distance

Focal Chord : A chord of a conic passing through its focus is called a focal chord .

Latus Rectum : The focal chord of a conic perpendicular to its axis is called is called its latus rectum.

PARABOLAProperties of Parabola :

1) The tangent at any point on the parabola bisects the angle between the focal chord through that point and a line perpendicular to the directrix

2) The intercept on the tangent of the parabola between its point of contact and the directrix subtends a right angle at the focus.

3) The sub-tangent is bisected at vertex

4) The circle on a focal radius of a parabola as diameter touches tangent at vertex .

5) The locus of the perpendiculars to the tangent of the parabola from the focus is tangent at vertex .

6) If the tangent at the point P and Q of parabola meet at K , then SK2 = SP. SQ

7) Through the vertex O of the parabola y2 = 4ax , a line is drawn perpendicular to any tangent , meeting it in P and the parabola in Q , then OP.OQ is constant.

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Tutorials '9892603993 MH-CET/XII Mathematics

PARABOLADefinition : A parabola is the locus of a point such that its distance from a fixed point is equal to

its distance from a fixed straight line.

The equation y2 = 4ax is called standard equation of parabola.

The equations x = at2 and y = 2at where ‘t’ the parameter , are called parametric equations

Parabola y2 = 4ax y2 = - 4ax x2= 4ay x2= -4ay

Focus (a, 0) (-a, 0) (0, a) (0, -a)

Directrix x + a = 0 x - a = 0 y + a = 0 y - a = 0

Axis

Equation of axis

X-axis

y = 0

X-axis

y = 0

Y –axis

x = 0

Y –axis

x = 0

Vertex (0, 0) origin (0, 0) origin (0, 0) origin (0, 0) origin

Length Latus Rectum 4a 4a 4a 4a

Ends of Latus Rectum (a, 2a) (a, -2a) (-a, 2a) (-a, -2a) (2a, a) (-2a, a) (2a, -a) (-2a, -a)

Equation of L. R. x - a = 0 x + a = 0 y - a = 0 y + a = 0

Tangent at vertex x = 0, Y –axis x = 0, Y –axis y = 0, X-axis y = 0, X-axis

Extent of Parabola I & IV quadrant II & III quadrant I & II quadrant III & IV quadrant

Standard Equation Parabola In general two tangents can be drawn to parabola from a point P( x1, y1) in its plane

If m1 and m2 are the slopes of the two tangents , then

If the tangents drawn at P(t) and Q(t) on the parabola intersect at R then

y2 = 4ax Parametric Equation x = at2 , y = 2at Equation of tangent at P(x1, y1) Equation of tangent at P ( ) yt = x + at2

Equation of tangent in terms of slope & Point of contact

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Tutorials '9892603993 MH-CET/XII Mathematics

E L L I P S E Ellipse ( a > b ) ( b > a )

Focus S( ae , 0 ) S (-ae, 0) S( 0 , be ) S (0, -be)

Directrix d , d d , d

Axis ( length )Major Axis (x-axis ) = 2aMinor Axis (y-axis ) = 2b

Major Axis (x-axis ) = 2bMinor Axis (y-axis ) = 2a

Centre ( 0, 0 ) ( 0, 0 )

Length Latus Rectum

Ends of Latus Rectum L L L L

Equation of L. R. x = ae , x = - ae y = be , y = - be

Relation between a & b b2 = a2 ( 1 –e2 ) a2 = b2 ( 1 –e2 )

VerticesA ( a, 0 ) , A’( -a, 0 )

B( 0, b ) , B’( 0, -b )

A ( a, 0 ) , A’( -a, 0 )

B( 0, b ) , B’( 0, -b )

Eccentricity

Distance between foci 2ae 2be

Distance between directrices

Standard EquationEllipse

Parametric Equation x = a cos , y = b sin Equation of tangent at P(x1, y1) Equation of tangent at P ( ) Equation of tangent in terms of slope & Point of contact Equation of Director Circle x2 + y2 = a2 + b2

H Y P E R B O L A

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Tutorials '9892603993 MH-CET/XII Mathematics

Hyperbola

Focus S( ae , 0 ) S (-ae, 0) S( 0 , be ) S (0, -be)

Directrix D , d d , d

Axis ( length )Transverse Axis (x-axis ) = 2aConjugate Axis (y-axis ) = 2b

Transverse Axis (x-axis ) = 2bConjugate Axis (y-axis ) = 2a

Centre ( 0, 0 ) ( 0, 0 )

Length Latus Rectum

Ends of Latus Rectum L L L L

Equation of L. R. x = ae , x = - ae y = be , y = - be

Relation between a & b b2 = a2 (e2 –1) a2 = b2 (e2 –1)

VerticesA ( a, 0 ) , A’( -a, 0 )

B( 0, b ) , B’( 0, -b )

A ( a, 0 ) , A’( -a, 0 )

B( 0, b ) , B’( 0, -b )

Eccentricity

Distance between foci 2ae 2be

Distance between directrices

Standard Equation

Hyperbola

Parametric Equation x = a sec , y = b tan Equation of tangent at P(x1, y1)

Equation of tangent at P ( )

Equation of tangent in terms of slope & Point of contact

Equation of Director Circle x2 + y2 = a2 – b2

Ellipse :

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Tutorials '9892603993 MH-CET/XII Mathematics

Definition : A ellipse is the locus of a point in a plane such that ratio of its distance from a fixed

point and fixed straight line is a constant less than one.

The equation is called standard equation of ellipse

The equations x = a cos and y = b sin where ‘ ’ the parameter, are called parametric

equations

In general two tangents can be drawn from a point P ( x1 , y1) to the ellipse

If m1 and m2 are slopes of these tangents then and

Properties of ellipse :

1) The sum of the focal distance of a point on the ellipse is equal to the length of the major axis.

OR The sum of the focal distance of a point on the ellipse is 2a

2) If p1 , p2 are the length of the perpendicular segments from the foci of the ellipse to

any tangent, then p1 . p2 = b2 OR

The product of the lengths of perpendicular segments from the foci on any tangent to the ellipse

is b2

3) The intercept on any tangent to an ellipse between its point of contact and the directrix

subtends right angle at the corresponding focus. OR

If the tangent at a point P on an ellipse meets a directrix in Q , then the segment PQ subtends

right angle at the corresponding focus

4) The focus of the foot of the perpendicular from a focus of the ellipse, on any

tangent is its auxiliary circle.

Hyperbola :

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Tutorials '9892603993 MH-CET/XII Mathematics

Definition : A hyperbola is the locus of a point in a plane such that ratio of its distance from a

fixed point and fixed straight line is a constant greater than one.

The equation is called standard equation of hyperbola

The equations x = a sec and y = b tan where ‘’ the parameter , are called parametric

equations

In general two tangents can be drawn from a point P ( x1 , y1) to the hyperbola

If m1 and m2 are slopes of these tangents then and

Properties of Hyperbola :

1) The difference between the focal distance of a point on the hyperbola is equal to the length of

its transverse axis. OR

If P is any point on the hyperbola , and S , S are the foci , then | SP – S P | = 2a .

2) The product of the lengths of the perpendicular from the foci and any tangent to the hyperbola

is equal to b2.

3) The tangent at a point P on a hyperbola bisects the angle between the focal radii to the point

P.

4) The locus of the foot of the perpendicular from the focus on the hyperbola on any

tangent is its auxiliary circle.

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