“Teach A Level Maths” Vol. 1: AS Core Modules 16: Circles © Christine Crisp

Module C1 Module C2 AQA Edexcel MEI/OCR OCR "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

P(x, y ) y x Consider a circle, with centre the origin and radius 1 Let P(x, y) be any point on the circle x y O P(x, y ) 1

Consider a circle, with centre the origin and radius 1 Let P(x, y) be any point on the circle By Pythagoras’ theorem for triangle OPM, x y O P(x, y ) 1 y M x

The equation gives a circle because only the coordinates of points on the circle satisfy the equation. e.g. Since the radius is 1, we can see that the point (1, 0) lies on the circle x y (1, 0) x 1

= the right hand side (r.h.s.) The equation gives a circle because only the coordinates of points on the circle satisfy the equation. e.g. Since the radius is 1, we can see that the point (1, 0) lies on the circle Substituting (1, 0) in the left hand side (l.h.s.) of the equation l.h.s. = the right hand side (r.h.s.) So, the equation is satisfied by the point (1, 0)

y x The point does not lie on the circle since l.h.s. (0. 5, 0. 5) x r.h.s. The equation is NOT satisfied by the point (0.5, 0.5). The point does not lie on the circle.

If we have a circle with centre at the origin but with radius r, we can again use Pythagoras’ theorem P(x, y ) x y O M We get r

Now consider a circle with centre at the point ( a, b ) and radius r. x y P(x, y ) y - b x x - a Using Pythagoras’ theorem as before:

Another way of finding the equation of a circle with centre ( a, b ) is to use a translation from x y x x Translate by : Replace x by (x – a) and y by (y – b)

SUMMARY The equation of a circle with centre ( a, b ) and radius r is We usually leave the equation in this form without multiplying out the brackets

Solution: Using the formula, e.g. Find the equation of the circle with centre ( 4, -3 ) and radius 5. Does the point ( 2, 1 ) lie on, inside, or outside the circle? Solution: Using the formula, the circle is ( 4 , -3 ) ( 2, 1 ) x Substituting the coordinates ( 2, 1 ): l.h.s. this gives the square of the distance of the point from the centre of the circle Since the distance of the point from the centre is less than the radius, the point ( 2, 1 ) is inside the circle

SUMMARY The equation of a circle with centre ( a, b ) and radius r is To determine whether a point lies on, inside, or outside a circle, substitute the coordinates of the point into the l.h.s. of the equation of the circle and compare the answer with

Exercises 1. Find the equation of the circle with centre (-1, 2 ) and radius 3. Multiply out the brackets to give your answer in the form Solution: Use a = -1, b = 2, r = 3 2. Determine whether the point (3,-5) lies on, inside or outside the circle with equation Solution: Substitute x = 3 and y = -5 in l.h.s. so the point lies outside the circle

Finding the centre and radius of a circle e.g. Find the centre and radius of the circle with equation Solution: First complete the square for x

Finding the centre and radius of a circle e.g. Find the centre and radius of the circle with equation Solution: First complete the square for x N.B. so we need to subtract 9 to get

Finding the centre and radius of a circle e.g. Find the centre and radius of the circle with equation Solution: First complete the square for x

Finding the centre and radius of a circle e.g. Find the centre and radius of the circle with equation Solution: Next complete the square for y

Finding the centre and radius of a circle e.g. Find the centre and radius of the circle with equation Solution: Copy the constant and complete the equation

Finding the centre and radius of a circle e.g. Find the centre and radius of the circle with equation Solution: Finally collect the constant terms onto the r.h.s. we can see the centre is ( 3, 2 ) and the radius is 5. By comparing with the equation ,

SUMMARY To find the centre and radius of a circle given in a form without brackets: Complete the square for the x-terms Complete the square for the y-terms Collect the constants on the r.h.s. Compare with The centre is (a, b) and the radius is r.

Exercises Find the centre and radius of the circle whose equation is (a) (b) Solution: Complete the square for x and y: Centre is ( 2, -4 ) and radius is 4 Solution: Complete the square for x and y: Centre is and radius is 3

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The equation of a circle with centre ( a, b ) and radius r is We usually leave the equation in this form without multiplying out the brackets SUMMARY To determine whether a point lies on, inside, or outside a circle, substitute the coordinates of the point into the l.h.s. of the equation of the circle and compare the answer with

Since the distance of the point from the centre is less than the radius, the point ( 2, 1 ) is inside the circle e.g. Find the equation of the circle with centre ( 4, -3 ) and radius 5. Does the point ( 2, 1 ) lie on, inside, or outside the circle? Substituting the coordinates ( 2, 1 ): l.h.s. Solution: Using the formula, the circle is

To find the centre and radius of a circle given in a form without brackets: Complete the square for the x-terms Complete the square for the y-terms Collect the constants on the r.h.s. Compare with The centre is (a, b) and the radius is r. SUMMARY

e.g. Find the centre and radius of the circle with equation Finally collect the constant terms onto the r.h.s. Solution: we can see the centre is ( 3, 2 ) and the radius is 5. By comparing with the equation ,