1. x2+y2x2+y2 x 2 +y 2 +Dx+Ey+F = 0x2+y2x2+y2 General Form x 2 +y 2 +Dx+Ey+F = 0 (a) How do we identify the equations of circles? The coefficients of x 2 and y 2 are both 1 No xy-terms (i) The coefficients of x 2 and y 2 must be the same; (ii) No xy-terms; (iii) The degree is 2. The degree is 2 13. Locus and Equations of Circles

2. Which of the following are equations of circles? (1) x 2 +y 2 +6x+14y+36 = 0E.g. (2) 5x 2 +5y 2 +25x － 45y+163 = 0 (4) x 2 +y 2 +18xy+68x － 19y － 78 = 0 (3) x 2 +6y 2 +18x － 45y － 64 = 0 (5) x 3 +y 2 +28x － 46y+85 = 0 13. Locus and Equations of Circles (a) How do we identify the equations of circles? (i) The coefficients of x 2 and y 2 must be the same; (ii) No xy-terms; (iii) The degree is 2.

3.  The coefficients of x 2 and y 2 are the same x2+y2x2+y2 (1) x 2 +y 2 +6x+14y+36 = 0x2+y2x2+y2  x 2 +y 2 +6x+14y+36 = 0 No xy-terms  The degree is 2  13. Locus and Equations of Circles (a) How do we identify the equations of circles? (i) The coefficients of x 2 and y 2 must be the same; (ii) No xy-terms; (iii) The degree is 2.

4. 5x 2 +5y 2 +25x － 45y+163 = 0 (2) 5x 2 +5y 2 +25x － 45y+163 = 0 5x 2 +5y 2  The coefficients of x 2 and y 2 are both 5  No xy-terms  The degree is 2  13. Locus and Equations of Circles (a) How do we identify the equations of circles? (i) The coefficients of x 2 and y 2 must be the same; (ii) No xy-terms; (iii) The degree is 2.

5. (3) x 2 +6y 2 +18x － 45y － 64 = 0  x 2 +6y 2 The coefficient of y 2 is not the same as that of x 2  13. Locus and Equations of Circles (a) How do we identify the equations of circles? (i) The coefficients of x 2 and y 2 must be the same; (ii) No xy-terms; (iii) The degree is 2.

6. x 2 +y 2 +18xy+68x － 19y － 78 = 0  (4) x 2 +y 2 +18xy+68x － 19y － 78 = 0 18xy There is a xy-term  13. Locus and Equations of Circles (a) How do we identify the equations of circles? (i) The coefficients of x 2 and y 2 must be the same; (ii) No xy-terms; (iii) The degree is 2.

7.  (5) x 3 +y 2 +28x － 46y+85 = 0 x3+y2x3+y2 The degree is 3  13. Locus and Equations of Circles (a) How do we identify the equations of circles? (i) The coefficients of x 2 and y 2 must be the same; (ii) No xy-terms; (iii) The degree is 2.

8. x 2 +y 2 +Dx+Ey+F+y2+y2 E 2 ( ) 2 + y + For the circle C: x 2 +y 2 +Dx+Ey+F = 0 (b) How do we distinguish between real circles, point circles and imaginary circles? E 2 ( ) 2 + E 2 ( ) 2 － D 2 ( ) 2 － x2 x2 +F+F x 2 +y 2 +Dx+Ey+F = 0 +Dx D 2 ( ) 2 + +Ey = 0 D 2 ( ) 2 + x = D 2 ( ) 2 ) － 2 E － F－ F The coordinates of the centre = E 2 ( ) 2 + D 2 －, ( r2r2 ∴ The radius = E 2 ( ) 2 + － F D 2 ( ) 2 E 2 ( ) 2 + E 2 ( ) 2 － D 2 ( ) 2 － D 2 ( ) 2 + 13. Locus and Equations of Circles

9. For the equation of a circle in general form, consider the value of the radius. (i) If then the circle is a real circle. E 2 ( ) 2 + － F D 2 ( ) 2 (iii) If then the circle is a point circle. (ii) If then the circle cannot be drawn and is called imaginary circle. 13. Locus and Equations of Circles (b) How do we distinguish between real circles, point circles and imaginary circles? E 2 ( ) 2 + － F ＞ 0, D 2 ( ) 2 E 2 ( ) 2 + － F = 0, D 2 ( ) 2 E 2 ( ) 2 + － F ＜ 0, D 2 ( ) 2

10. Determine the type of circle that each of the following (2) x 2 +y 2 － 16x － 8y+80 = 0 (1) x 2 +y 2 +8x+6y+15 = 0 (3) x 2 +y 2 +14x+4y+60 = 0 equations represents. 13. Locus and Equations of Circles (b) How do we distinguish between real circles, point circles and imaginary circles?E.g.

11. x 2 +y 2 +Dx+Ey+F = 0 E 2 ( ) 2 + － F D 2 ( ) 2 10 = 0 ＞ ∴ It is a real circle (1) x 2 +y 2 +8x+6y+15 = 0 13. Locus and Equations of Circles (b) How do we distinguish between real circles, point circles and imaginary circles? = (4) 2 +(3) 2 － 15 6 2 ( ) 2 + － 15 8 2 ( ) 2 =

12. = ( － 8) 2 +( － 4) 2 － 80 x 2 +y 2 +Dx+Ey+F = 0 －8－8 2 ( ) 2 + － 80 － 16 2 ( ) 2 = 0 = ∴ It is a point circle (2) x 2 +y 2 － 16x － 8y+80 = 0 E 2 ( ) 2 + － F D 2 ( ) 2 13. Locus and Equations of Circles (b) How do we distinguish between real circles, point circles and imaginary circles?

13. = (7) 2 +(2) 2 － 60 x 2 +y 2 +Dx+Ey+F = 0 －7－7 = 0 ＜ ∴ It is an imaginary circle (3) x 2 +y 2 +14x+4y+60 = 0 E 2 ( ) 2 + － F D 2 ( ) 2 13. Locus and Equations of Circles (b) How do we distinguish between real circles, point circles and imaginary circles? 4 2 ( ) 2 + － 6060 14 2 ( ) 2 =