Circles © Christine Crisp Objectives To know the equation of a circle (Cartesian form) To find the intersection of circles with straight lines To Find the tangent to a circle To know three circle theorems To solve circle problems using these theorems Keywords Chord, Tangent, bisector, perpendicular, gradient, semi-circle

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

The equation of a 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 x y By Pythagoras’ theorem for triangle OPM, M

The equation of a circle x y 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 (1, 0) x 1

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

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

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

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

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

The equation of a circle 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

The equation of a circle 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 this gives the square of the distance of the point from the centre of the circle ( 4, -3 ) ( 2, 1 ) x x

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

The equation of a circle Use 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 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 Solution: a =  1, b = 2, r = 3

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

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

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

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

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

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

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.