Find the equation of the circle with centre 1 2 , - 3 2 and radius 5 2 .

Equation of circle with centre ( a , b ) and radius r is given by ( x - a ) 2 + ( y - b ) 2 = r 2 ∴ Equation of circle with centre 1 2 , - 3 2 and radius 5 2 is x - 1 2 2 + y + 3 2 2 = 25 4 ⇒ x 2 - x + 1 4 + y 2 + 3 y + 9 4 = 25 4 ⇒ x 2 + y 2 - x + 3 y - 15 4 = 0 4 x 2 + 4 y 2 - 4 x + 12 y - 15 = 0

A circle passes through the points ( 3 , 2 ) and ( 7 , 2 ) and has radius 2 2 . Find the two possible equations for this circle.

Given points are ( 3 , 2 ) and ( 7 , 2 ) If r = 2 2 , then r 2 = 8 Equation of the circle is ( x - a ) 2 + ( y - b ) 2 = r 2 Substituting x = 3 , y = 2 and r 2 = 8 ⇒ ( 3 - a ) 2 + ( 2 - b ) 2 = 8 ......(i) Substituting x = 7 , y = 2 and r 2 = 8 ⇒ ( 7 - a ) 2 + ( 2 - b ) 2 = 8 ........(ii) (i) - (ii) ⇒ ( 3 - a ) 2 - ( 7 - a ) 2 = 0 ⇒ 9 - 6 a + a 2 - 49 - 14 a + a 2 = 0 ⇒ 9 - 6 a + a 2 - 49 + 14 a - a 2 = 0 ⇒ 8 a - 40 = 0 ⇒ a = 5 From (i), ( 3 - 5 ) 2 + ( 2 - b ) 2 = 8 ⇒ 4 + ( 2 - b ) 2 = 8 ⇒ ( 2 - b ) 2 = 4 ⇒ 2 - b = ± 2 ⇒ b = 0 or b = 4 Substituting a = 5 , b = 0 and r 2 = 8 into ( x - a ) 2 + ( y - b ) 2 = r 2 ⇒ ( x - 5 ) 2 + ( y - 0 ) 2 = 8 ⇒ ( x - 5 ) 2 + y 2 = 8 Substituting a = 5 , b = 4 and r 2 = 8 into ( x - a ) 2 + ( y - b ) 2 = r 2 ⇒ ( x - 5 ) 2 + ( y - 4 ) 2 = 8 ∴ The two possible equations for the circle ( x - 5 ) 2 + y 2 = 8 and ( x - 5 ) 2 + ( y - 4 ) 2 = 8

E M B I B E

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3D>Q 2

EASY

AS and A Level

IMPORTANT

Earn 100

Find the equation of the circle with centre $(- 1, 3)$ and radius $\sqrt{7}$ .

Important Questions on Coordinate Geometry

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3D>Q 2

Find the equation of the circle with centre $(\frac{1}{2}, - \frac{3}{2})$ and radius $\frac{5}{2}$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3D>Q 3

Find the equation of the circle with centre $(2, 5)$ passing through the point $(6, 8)$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3D>Q 4

A diameter of a circle has its end points at $A (- 6, 8)$ and $B (2, - 4)$ . Find the equation of the circle.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3D>Q 5

Sketch the circle $(x - 3)^{2} + (y + 2)^{2} = 9$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3D>Q 6

Find the equation of the circle that touches the $x -$ axis and whose centre is $(6, - 5)$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3D>Q 7

The points $P (1, - 2)$ and $Q (7, 1)$ lie on the circumference of a circle. Show that the centre of the circle lies on the line $4 x + 2 y = 15$ .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3D>Q 8

A circle passes through the points $(3, 2)$ and $(7, 2)$ and has radius $2 \sqrt{2}$ . Find the two possible equations for this circle.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 3 - Coordinate Geometry>EXERCISE 3D>Q 9

A circle passes through the points $O (0, 0), A (8, 4)$ and $B (6, 6)$ . Show that $O A$ is a diameter of the circle and find the equation of this circle.