1. HigherUnit 2 Outcome 4 Circle x 2 + y 2 + 2gx + 2fy + c = 0 The General equation of a circle Wednesday, 07 January 2009

2. Higher Circle Unit 2 Outcome 4 x 2 + y 2 + 2gx + 2fy + c = 0 The equation of a circle is (x – 2) 2 + (y – 3) 2 = 25 Write the equation without brackets (x – 2) (x – 2) + (y – 3) (y – 3) = 25 x 2 - 4x + 4 + y 2 - 6y + 9 = 25 x 2 - 4x + y 2 - 6y + 13 - 25 = 0 x 2 + y 2 - 4x - 6y - 12 = 0 x 2 + y 2 – 2ax – 2by +a 2 +b 2 – r 2 = 0 (x – a) (x – a) + (y – b) (y – b) = r 2 (x – a) 2 + (y – b) 2 = r 2 x 2 – 2ax + a 2 + y 2 - 2by + b 2 = r 2 As a, b and r are constants (numbers) then these can be collected together as one term, c x 2 + y 2 – 2ax – 2by + c = 0 In the same way we can This is the general form of the equation of a circle Wednesday, 07 January 2009

3. Centre C(a,b) Radius r 1. Radius r Centre C(-g,-f) 2. x 2 + y 2 + 2gx + 2fy + c = 0 HigherUnit 2 Outcome 4 Circle Wednesday, 07 January 2009

4. Finding the centre and the radius Given the equation of a circle, we can find the coordinates of its centre and the length of its radius. For example: Find the centre and the radius of a circle with the equation ( x – 2) 2 + ( y + 7) 2 = 64 By comparing this to the general form of the equation of a circle of radius r centred on the point ( a, b ): ( x – a ) 2 + ( y – b ) 2 = r 2 We can deduce that for the circle with equation ( x – 2) 2 + ( y + 7) 2 = 64 The centre is at the point (2, –7) and the radius is 8. Wednesday, 07 January 2009

5. Finding the centre and the radius When the equation of a circle is given in the form Find the centre and the radius of a circle with the equation x 2 + y 2 + 4 x – 6 y + 9 = 0 Start by rearranging the equation so that the x terms and the y terms are together: x 2 + 4 x + y 2 – 6 y + 9 = 0 x 2 + y 2 – 2 ax – 2 by + c = 0 we can use the method of completing the square to write it in the form ( x – a ) 2 + ( y – b ) 2 = r 2 For example: Wednesday, 07 January 2009

6. Finding the centre and the radius We can complete the square for the x terms and then for the y terms as follows: The equation of the circle can now be written as: x 2 + 4 x =( x + 2) 2 – 4 y 2 – 6 y =( y – 3) 2 – 9 ( x + 2) 2 – 4 + ( y – 3) 2 – 9 + 9 = 0 ( x + 2) 2 + ( y – 3) 2 = 4 ( x + 2) 2 + ( y – 3) 2 = 2 2 The centre is at the point (–2, 3) and the radius is 2. x 2 + 4 x + y 2 – 6 y + 9 = 0 Wednesday, 07 January 2009

7. Higher Circle Unit 2 Outcome 4 x 2 + y 2 + 2gx + 2fy + c = 0 Alternative approach Rearrange to get in the general form x 2 + 4 x + y 2 – 6 y + 9 = 0 x 2 + y 2 + 4x – 6 y + 9 = 0 2g = 4 2f = -6 c = 9 x 2 + y 2 + 2gx + 2fy + c = 0 g = 2 f = -3 c = 9 As before It therefore follows that The centre is at the point (–2, 3) and the radius is 2. ( x + 2) 2 + ( y – 3) 2 = 2 2 C is sum of all the constants Wednesday, 07 January 2009 r 2 = g 2 +f 2 - c r 2 = 2 2 + - 3 2 - 9 Centre (-g, -f)

8. Higher Circle Unit 2 Outcome 4 x 2 + y 2 + 2gx + 2fy + c = 0 Wednesday, 06 January 2009 Show that the equation x 2 + y 2 - 6x + 2y - 71 = 0 represents a circle and find the centre and radius. x 2 + y 2 - 6x + 2y - 71 = 0 2g = -6 2f = 2 c = -71 g = -3 f = 1 c = -71 ( x + 3) 2 + ( y – 1) 2 = 9 2 r 2 = g 2 + f 2 -c r 2 = 9 + 1 - -71 r 2 = 81 This is now in the form (x-a) 2 + (y-b) 2 = r 2 So represents a circle with centre (3,-1) and radius = 9 Centre (-g, -f)

9. Higher Circle Unit 2 Outcome 4 x 2 + y 2 + 2gx + 2fy + c = 0 Wednesday, 06 January 2009 Show that the equation x 2 + y 2 + 6x - 2y - 15 = 0 represents a circle and find the centre and radius. x 2 + y 2 + 6x - 2y - 15 = 0 2g = 6 2f = -2 c = -15 g = 3 f = -1 c = -15 ( x - 3) 2 + ( y + 1) 2 = 5 2 r 2 = g 2 + f 2 -c r 2 = 9 + 1 - -15 r 2 = 25 This is now in the form (x-a) 2 + (y-b) 2 = r 2 So represents a circle with centre (-3,1) and radius = 5 Centre (-g, -f)

10. Higher Circle Unit 2 Outcome 4 x 2 + y 2 + 2gx + 2fy + c = 0 Wednesday, 06 January 2009 Show that the equation x 2 + y 2 - 4x - 6y + 9 = 0 represents a circle and find the centre and radius. x 2 + y 2 - 4x - 6y + 9 = 0 2g = -4 2f = -6 c = 9 g = -2 f = -3 c = 9 ( x + 2) 2 + ( y + 3) 2 = 2 2 r 2 = g 2 + f 2 -c r 2 = 4 + 9 - 9 r 2 = 4 This is now in the form (x-a) 2 + (y-b) 2 = r 2 So represents a circle with centre (2,3) and radius = 2 Centre (-g, -f)

11. Higher Circle Unit 2 Outcome 4 x 2 + y 2 + 2gx + 2fy + c = 0 Wednesday, 06 January 2009 Show that the equation x 2 + y 2 + 2x + 8y + 1 = 0 represents a circle and find the centre and radius. x 2 + y 2 + 2x + 8y + 1 = 0 2g = 2 2f = 8 c = 1 g = 1 f = 4 c = 1 ( x - 1) 2 + ( y - 4) 2 = 4 2 r 2 = g 2 + f 2 - c r 2 = 1 + 16 -1 r 2 = 16 This is now in the form (x-a) 2 + (y-b) 2 = r 2 So represents a circle with centre (-1,-4) and radius = 4 Centre (-g, -f)

12. HigherUnit 2 Outcome 4 Circle Wednesday, 06 January 2009 Page 172 To build skills Complete Exercise 3A Q 1, Q2, (x – a) 2 + (y – b) 2 = r 2 (a,b)r Centre C (a,b) and radius r

13. HigherUnit 2 Outcome 4 Circle Tuesday, 06 January 2009 What do you see ?