Teach A Level Maths Vol 1 AS Core
- Slides: 28
“Teach A Level Maths” Vol. 1: AS Core Modules 16: Circles © Christine Crisp
The equation of a circle Module C 1 Module C 2 AQA Edexcel MEI/OCR "Certain images and/or photos on this presentation are the copyrighted property of Jupiter. Images and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from Jupiter. Images"
The equation of a circle Consider a circle, with centre the origin and radius 1 Let P(x, y) be any point on the circle y P(x, y ) 1 O x
The equation of a circle 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, y P(x, y ) 1 O x y M x
The equation of a circle 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 y 1 x (1, 0) x
The equation of a circle 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)
The equation of a circle The point does not lie on the circle since y l. h. s. (0. 5, 0. 5) x x r. h. s. The equation is NOT satisfied by the point (0. 5, 0. 5). The point does not lie on the circle.
The equation of a circle If we have a circle with centre at the origin but with radius r, we can again use Pythagoras’ theorem We get y P(x, y ) r O y x M x
The equation of a circle Now consider a circle with centre at the point ( a, b ) and radius r. y P(x, y ) y-b x x-a x Using Pythagoras’ theorem as before:
The equation of a circle Another way of finding the equation of a circle with centre ( a, b ) is to use a translation from y x x Replace x by (x – a) and y by • Translate by (y – b) : x
The equation of a circle 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
The equation of a 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? Solution: Using the formula, the circle is Substituting the coordinates ( 2, 1 ): l. h. s. x ( 2, 1 ) x( 4 , -3 ) 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
The equation of a 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 The equation of a circle 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
The equation of a 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
The equation of a 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 N. B. so we need to subtract 9 to get
The equation of a 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
The equation of a circle 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
The equation of a circle 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
The equation of a circle 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. By comparing with the equation we can see the centre is ( 3, 2 ) and the radius is 5. ,
The equation of a circle 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.
The equation of a circle 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
The equation of a circle
The equation of a circle The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
The equation of a circle SUMMARY Ø 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 Ø 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
The equation of a 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? Solution: Using the formula, the circle is Substituting the coordinates ( 2, 1 ): l. h. s. Since the distance of the point from the centre is less than the radius, the point ( 2, 1 ) is inside the circle
The equation of a circle 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.
The equation 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. By comparing with the equation , see the centre is ( 3, 2 ) and the radius is 5. we can