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x 2 + y 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. x 2 + y 2. x 2 + y 2. x 2 + y 2 + Dx+Ey+F = 0. General Form x 2 + y 2 + Dx+Ey+F = 0.

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x 2 + y 2

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  1. 13. Locus and Equations of Circles (a) How do we identify the equations of circles? (i) The coefficients of x2 and y2 must be the same; (ii) No xy-terms; (iii) The degree is 2. x2+y2 x2+y2 x2+y2+Dx+Ey+F= 0 General Formx2+y2+Dx+Ey+F= 0 No xy-terms The degree is 2 The coefficients of x2and y2are both 1

  2. E.g. 13. Locus and Equations of Circles (a) How do we identify the equations of circles? (i) The coefficients of x2 and y2 must be the same; (ii) No xy-terms; (iii) The degree is 2. Which of the following are equations of circles? (1) x2+y2+6x+14y+36= 0 (2) 5x2+5y2+25x－45y+163= 0 (3) x2+6y2+18x－45y－64= 0 (4) x2+y2+18xy+68x－19y－78= 0 (5) x3+y2+28x－46y+85= 0

  3. 13. Locus and Equations of Circles (a) How do we identify the equations of circles?  (i) The coefficients of x2 and y2 must be the same; (ii) No xy-terms; (iii) The degree is 2.    (1) x2+y2+6x+14y+36= 0 x2+y2 x2+y2 x2+y2+6x+14y+36= 0 Thecoefficients ofx2and y2 are the same No xy-terms The degree is 2

  4. 13. Locus and Equations of Circles (a) How do we identify the equations of circles?  (i) The coefficients of x2 and y2 must be the same; (ii) No xy-terms; (iii) The degree is 2.    (2) 5x2+5y2+25x－45y+163 = 0 5x2+5y2 5x2+5y2 5x2+5y2+25x－45y+163 = 0 Thecoefficients ofx2and y2 are both 5 No xy-terms The degree is 2

  5. 13. Locus and Equations of Circles (a) How do we identify the equations of circles?  (i) The coefficients of x2 and y2 must be the same; (ii) No xy-terms; (iii) The degree is 2.  (3) x2+6y2+18x－45y－64= 0 x2+6y2 The coefficient ofy2is not the same as that of x2

  6. 13. Locus and Equations of Circles (a) How do we identify the equations of circles? (i) The coefficients of x2 and y2 must be the same; (ii) No xy-terms; (iii) The degree is 2.   (4) x2+y2+18xy+68x－19y－78= 0 x2+y2+18xy+68x－19y－78= 0 18xy There is a xy-term

  7. 13. Locus and Equations of Circles (a) How do we identify the equations of circles? (i) The coefficients of x2 and y2 must be the same; (ii) No xy-terms; (iii) The degree is 2.   (5) x3+y2+28x－46y+85= 0 x3+y2 The degree is 3

  8. D D 2 2 ( ( + + ) ) 2 2 E E 2 2 ( ( + + ) ) 2 2 2 2 E E － － ( ( ) ) 2 2 D D 2 2 － － ( ( ) ) 2 2 13. Locus and Equations of Circles (b) How do we distinguish between real circles, point circles and imaginary circles? For the circle C: x2+y2+Dx+Ey+F = 0 x2+y2+Dx+Ey+F = 0 x2 x2+y2+Dx+Ey+F +y2 +Dx +Ey +F = 0 2 2 D E D D D E E E 2 2 2 2 x y ( ( + ( + ( ( ( ( － F － ) ) ) ) ) ) , － ) + + + －F = 2 2 2 2 2 2 2 2 r2 The coordinates of the centre = ∴The radius =

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

  10. E.g. 13. Locus and Equations of Circles (b) How do we distinguish between real circles, point circles and imaginary circles? Determinethe type of circle that eachof the following equations represents. (1) x2+y2+8x+6y+15= 0 (2) x2+y2－16x－8y+80= 0 (3) x2+y2+14x+4y+60= 0

  11. 8 6 2 2 ( ) + ( － 15 = ) 2 2 13. Locus and Equations of Circles (b) How do we distinguish between real circles, point circles and imaginary circles? ∴Itis a real circle (1) x2+y2+8x+6y+15= 0 x2+y2+Dx+Ey+F= 0 E D 2 2 + ( ( － F ) ) 2 2 10 = ＞ 0 = (4)2+(3)2－15

  12. －16 －8 2 2 = ( ) + ( － 80 ) 2 2 0 = 13. Locus and Equations of Circles (b) How do we distinguish between real circles, point circles and imaginary circles? ∴It is a point circle (2) x2+y2－16x－8y+80= 0 x2+y2+Dx+Ey+F= 0 E D 2 2 + ( ( － F ) ) 2 2 = (－8)2+(－4)2－80

  13. 14 4 2 2 = ( ) + ( － 60 ) 2 2 －7 = 13. Locus and Equations of Circles (b) How do we distinguish between real circles, point circles and imaginary circles? ∴It is an imaginary circle (3) x2+y2+14x+4y+60= 0 x2+y2+Dx+Ey+F= 0 D E 2 2 + ( ( － F ) ) 2 2 ＜ 0 = (7)2+(2)2－60

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