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Understanding the General Equation of a Circle and Finding Center and Radius

This guide elaborates on the general equation of a circle expressed as ( x^2 + y^2 + 2gx + 2fy + c = 0 ). It explains how to derive the center and radius of a circle from its equation using different forms, such as ( (x-a)^2 + (y-b)^2 = r^2 ). The document includes examples, methods for completing the square, and step-by-step processes for finding the coordinates of the center (C) and the length of the radius (r) for given equations. Perfect for students learning conic sections.

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Understanding the General Equation of a Circle and Finding Center and Radius

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

  2. Higher Circle Unit 2 Outcome 4 x 2 + y 2 + 2gx + 2fy + c = 0 In the same way we can The equation of a circle is (x – 2)2 + (y – 3)2 = 25 Write the equation without brackets (x – a) 2 + (y – b) 2 = r2 (x – a)(x – a) + (y – b)(y – b) = r2 (x – 2)(x – 2) + (y – 3)(y – 3) = 25 x2 – 2ax + a2 + y2 - 2by + b2 = r2 x2 - 4x + 4 + y2 - 6y + 9 = 25 x2 + y2 – 2ax – 2by +a2 +b2 – r2 = 0 x2 - 4x + y2 - 6y + 13 - 25 = 0 As a , b and r are constants (numbers) then these can be collected together as one term, c x2 + y2 - 4x - 6y - 12 = 0 x2 + y2 – 2ax – 2by + c = 0 This is the general form of the equation of a circle Wednesday, 07 January 2009

  3. Higher Circle Unit 2 Outcome 4 Radiusr 2. Centre C(-g,-f) x 2 + y 2 + 2gx + 2fy + c = 0 Radiusr 1. Centre C(a,b) 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 = r2 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 equationof a circle is given in the form x2+ y2– 2ax – 2by + c= 0 we can use the method of completing the square to write it in the form (x– a)2 + (y– b)2 = r2 For example: Find the centre and the radius of a circle with the equation x2+ y2+ 4x – 6y + 9= 0 Start by rearranging the equation so that the x terms and the y terms are together: x2+ 4x+y2– 6y + 9= 0 Wednesday, 07 January 2009

  6. Finding the centre and the radius x2+ 4x+y2– 6y + 9= 0 We can complete the square for the x terms and then for the y terms as follows: x2 + 4x = (x + 2)2– 4 y2– 6y = (y– 3)2– 9 The equation of the circle can now be written as: (x + 2)2– 4 + (y– 3)2– 9 + 9 = 0 (x + 2)2 + (y– 3)2= 4 (x + 2)2 + (y– 3)2= 22 The centre is at the point (–2, 3) and the radius is 2. Wednesday, 07 January 2009

  7. Higher Circle Unit 2 Outcome 4 x 2 + y 2 + 2gx + 2fy + c = 0 Alternative approach x2+ 4x+y2– 6y + 9= 0 Rearrange to get in the general form C is sum of all the constants x 2 + y 2 + 2gx + 2fy + c = 0 x2+y2 + 4x – 6y + 9= 0 2g= 4 2f= -6 c= 9 g= 2 f= -3 c= 9 r2 = 22 + - 32 - 9 (x + 2)2 + (y– 3)2= 22 r2 = g2 +f2 - c As before It therefore follows that Centre (-g, -f) The centre is at the point (–2, 3) and the radius is 2. Wednesday, 07 January 2009

  8. Higher Circle Unit 2 Outcome 4 x 2 + y 2 + 2gx + 2fy + c = 0 Show that the equation x2 + y2 - 6x + 2y - 71 = 0 represents a circle and find the centre and radius. x2 + y2 - 6x + 2y - 71 = 0 2g= -6 2f= 2 c= -71 r2 = g2 + f2 -c c= -71 g= -3 f= 1 r2 = 9 + 1 - -71 (x + 3)2 + (y– 1)2= 92 r2 = 81 This is now in the form (x-a)2 + (y-b)2 = r2 Centre (-g, -f) So represents a circle with centre (3,-1) and radius = 9 Wednesday, 06 January 2009

  9. Higher Circle Unit 2 Outcome 4 x 2 + y 2 + 2gx + 2fy + c = 0 Show that the equation x2 + y2 + 6x - 2y - 15 = 0 represents a circle and find the centre and radius. x2 + y2 + 6x - 2y - 15 = 0 2g= 6 2f= -2 c= -15 r2 = g2 + f2 -c c= -15 g= 3 f= -1 r2 = 9 + 1 - -15 (x - 3)2 + (y+ 1)2= 52 r2 = 25 Centre (-g, -f) This is now in the form (x-a)2 + (y-b)2 = r2 So represents a circle with centre (-3,1) and radius = 5 Wednesday, 06 January 2009

  10. Higher Circle Unit 2 Outcome 4 x 2 + y 2 + 2gx + 2fy + c = 0 Show that the equation x2 + y2 - 4x - 6y + 9 = 0 represents a circle and find the centre and radius. x2 + y2 - 4x - 6y + 9 = 0 2g= -4 2f= -6 c= 9 r2 = g2 + f2 -c c= 9 g= -2 f= -3 r2 = 4 + 9 - 9 (x + 2)2 + (y+ 3)2= 22 r2 = 4 Centre (-g, -f) This is now in the form (x-a)2 + (y-b)2 = r2 So represents a circle with centre (2,3) and radius = 2 Wednesday, 06 January 2009

  11. Higher Circle Unit 2 Outcome 4 x 2 + y 2 + 2gx + 2fy + c = 0 Show that the equation x2 + y2 + 2x + 8y + 1 = 0 represents a circle and find the centre and radius. x2 + y2 + 2x + 8y + 1 = 0 2g= 2 2f= 8 c= 1 r2 = g2 + f2 - c c= 1 g= 1 f= 4 r2 = 1 + 16 -1 (x - 1)2 + (y- 4)2= 42 r2 = 16 Centre (-g, -f) This is now in the form (x-a)2 + (y-b)2 = r2 So represents a circle with centre (-1,-4) and radius = 4 Wednesday, 06 January 2009

  12. Higher Circle Unit 2 Outcome 4 Centre C(a,b)and radiusr (x – a)2 + (y – b)2 = r2 Page 170 To build skills Complete Exercise 3A Q 1, Q2, Wednesday, 06 January 2009

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

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