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OBJECTIVES :

4.1 CIRCLES

(a) Determine the equation of a circle.(b) Determine the centre and radius of a

circle by completing the square(c) Find the points of intersection of two

circles(d) Find the equation of tangents and

normal to a circle(e) Find the length of a tangent from a point to a circle

4.0 CONIC SECTIONS

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A circle is a set of all points in a plane equidistant from a given fixed pointcalled the center.

The distance from the center to any

point on the circle is called a radius.

Definition

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The Equation of a Circle in Standard Form

1) Center (0,0) , radius : r

Equation of a circle : 222 ryx

(0,0)

P(x , y)r

Proof: OP

2OP x

y

22 00 yx

2r 2 2x y

2 2x y

If a center of the circle is not located at

the origin but at any arbitrary point

( h,k ), the equation becomes :

2 2 2x h y k r

x

y

(h,k)

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2) Center ( h , k ) , radius ; r

Equation of a circle :

C (h , k)

r

y

x

Proof:

22 kyhx

P (x,y) r CP

222 rkyhx

222 kyhxr

From the Standard Equation,

2 2 2 2 22 2 0x y hx ky h k r

2 2 2( ) ( ) ..........(1)x h y k r

Expand and rearrange equation (1),

Then substitute ;2 2 2, ,h g k f c h k r

We get General Equation, 2 2 2 2 0x y gx fy c

The Equation of a Circle in General Form

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From the General Equation2 2 2 2 0x y gx fy c

By completing the square;2 2 2 2( ) ( ) 0x g y f g f c 2 2( ) ( )x g y f 2 2g f c

Centre , ( , )C g f

Radius , 2 2r g f c

By comparing to Standard Equation

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2 2 2( ) ( )x h y k r

( , )h k

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Find the general equation of the circle with centre (-2,3) and radius 5 .

222 rkyhx

Example 1Example 1

SolutionSolution

Determine the center and radius of a circle

by completing the

square.

0446 22 yyxx

Example 2Example 2

SolutionSolution

; (3, 2) 3ans c r

Find the center of the circle and the length of it’s radius

2 22 2 28 12 114 0x y x y

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Example 3 Example 3

SolutionSolution

2 2 14 6 57 0x y x y

2 22 2 28 12 114 0x y x y 2

02222 cfygxyx

compare with general equation :

; : g, f (7, 3) , 1ans centre r

Find the general equation of the circle with center (0,7) and touches the line 3x = 32 + 4y

Example 4 Example 4

SolutionSolution

C (0 ,7)

r

3 4 32 0x y

d r

2 2

ax by cd

a b

Shortest distance = d

2 2; 14 95 0ans x y y

Given that a circle passes through (9, -7) ,

(-3, -1) and (6,2 ). Find its equation.

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Example 5 Example 5

SolutionSolution

General equation of circle2 2 2 2 0x xy yg f c

2 2; 6 8 20 0ans x y x y

Find the equation of circle passing through the points (1,1), (3,2) and with the equation of diameter y-3x+7 = 0.

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Example 6 Example 6

SolutionSolution

(1,1)

(3,2)

General equation :2 2 2 2 0x y gx fy c

(1)At (1,1) : 2 2 2 0g f c

(2)At (3,2) : 13 6 4 0g f c

y–3x+7=0

C(-g,-f)

(1,1)

(3,2)

The diameter must passesthrough the centre,(-g ,-f )

(3)3 7 0f g

2 2; 5 4 0ans x y x y

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The points A and B have coordinates ( x1,

y1) and (x2 , y2). Show that the equation

of a circle where AB is the diameter of a circle is 1 2 1 2 0x x x x y y y y

Hence , find the equation of the circle

where AB is a diameter and the points of

A and B are ( 1 , 1 ) and ( 2 , 3) .

Example 7 Example 7

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A (x1,y1) B (x2, y2)

P (x, y)

AP and PB is perpendicular :

AP PBm m 1

APm PBm 1

1

,y y

x x

2

2

y y

x x

SolutionSolution

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A (x1,y1) B (x2,y2)

P (x,y)

1AP PBm m

2 2equation of circle; 4 3 5 0x y y x

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Important Notes

Given the general equation of two circles: If 2( ) 4 0a b ac Two circles

intersect at two distinct points

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2( ) 4 0b b ac Two circles touch to each other

2( ) 4 0c b ac Two circles do not intersect to each other

Find the intersection points between the two circles below:

2 2 2 24 and 4 3 0x y x y x

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Example 8 Example 8

....... (1)

...... (2)

2 2 4x y

2 2 4 3 0x y x

SolutionSolution

7 15 7 15; , and ,

4 4 4 4ans

2 2Show that the circles 2 4 4 andx y x y 2 2 8 10 32 0 intersect at two

distinct points.

Hence, find the intersection points.

x y x y

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Example 9 Example 9

2 2 2 4 4 x y x y 2 2 8 10 32 0x y x y

....... (1)

....... (2)

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SolutionSolution

; (1,5) and (4,2)ans

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(1) Standard equation : x2 + y2 = r2

The equation of tangents to a circle at

the point P ( x1, y1) is given by

xx1 + yy1 = r2

0

P(x1 , y1)

x

y

The equations of tangents and normal to a circle

xx1 + yy1 = r2

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(2) General equation :

xx1 + yy1 + g (x + x1) + f (y + y1) +c = 0

2 2 2 2 0x y gx fy c

The equation of tangents to a circle at

the point P ( x1, y1) is given by

Find tangent and normal line to the circlex2 + y2 = 5 at the point ( -2, 1).

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Example 10 Example 10

SolutionSolution

The equation of tangent at ( -2,1)

xx1 + yy1 = r2

-2x + y(1) = 5

y = 2x + 5

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31

2tm

The equation of normal at ( -2,1)

1y

nm

2x 1

2

1

2

1 1( )ny y m x x

2 2 ( 2)y x

2 0x y

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Find the equation of the tangent and normal of the circle at the point (3,1)

2 2 8 6 8 0x y x y

Example 11 Example 11

SolutionSolution

2 2 2 2 0x y gx fy c

2 2 8 6 8 0x y x y

Equation of tangent at (3,1),

; 4 1 0ans y x

Equation of normal at (3,1),

; 4 7ans y x

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TheoremThe length of the tangent from a fixed point P(x1, y1) to a circle with equation

x2 + y2 +2gx + 2fy+ c = 0 is given by

C

r

P (x1,y1)

d

Q

The length of a tangent from a point to a circle

2 21 1 1 12 2PQ d x y g x f y c

Find the length of the tangent from the point A (6,7) to a circle x2 + y2 – 2x – 8y = 8

Example 12 Example 12

SolutionSolution

; 3ans