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...

x2+y2x2+y2+Dx+Ey+F = 0x2+y2General Form x2+y2+Dx+Ey+F = 0

(a) How do we identify the equations of circles?

The coefficients of x2

and y2 are both 1No xy-terms

(i) The coefficients of x2 and y2 must be the same;(ii) No xy-terms;(iii) The degree is 2.

The degree is 2

13. Locus and Equations of Circles13. Locus and Equations of Circles

Which of the following are equations of circles?

(1) x2+y2+6x+14y+36 = 0

E.gE.g..

(2) 5x2+5y2+25x － 45y+163 = 0

(4) x2+y2+18xy+68x － 19y － 78 = 0

(3) x2+6y2+18x － 45y － 64 = 0

(5) x3+y2+28x － 46y+85 = 0

13. Locus and Equations of Circles13. 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.

The coefficients of x2 and y2 are the same

x2+y2(1) x2+y2+6x+14y+36 = 0x2+y2

x2+y2+6x+14y+36 = 0

No xy-terms

The degree is 2

13. Locus and Equations of Circles13. 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.

5x2+5y2+25x － 45y+163 = 0(2) 5x2+5y2+25x － 45y+163 = 05x2+5y25x2+5y2The coefficients of x2 and y2 are both 5

No xy-terms

The degree is 2

13. Locus and Equations of Circles13. 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 of y2 is not the same as that of x2

13. Locus and Equations of Circles13. 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+18xy+68x － 19y － 78 = 0 (4) x2+y2+18xy+68x － 19y － 78 = 0

18xy

There is a xy-term

13. Locus and Equations of Circles13. 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 = 0x3+y2

The degree is 3

13. Locus and Equations of Circles13. 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+Dx+Ey+F +y2

E2

( )2

+y+

For the circle C: x2+y2+Dx+Ey+F = 0

(b) How do we distinguish between real circles, point circles and imaginary circles?

E2

( )2

+ E2

( )2

－D2

( )2

－ x2 +F

x2+y2+Dx+Ey+F = 0

+Dx D2

( )2

++Ey = 0

D2

( )2

+x = D2

( )2

)－ 2E

－ F

The coordinates of the centre =

E2

( )2

+

D2－ ,(

r2

∴ The radius = E2

( )2

+ － FD2

( )2

E2

( )2

+ E2

( )2

－D2

( )2

－D2

( )2

+

13. Locus and Equations of Circles13. Locus and Equations of Circles

For the equation of a circle in general form,

consider the value of the radius .

(i) If

then the circle is a real circle.

E2

( )2

+ － FD2

( )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 Circles13. Locus and Equations of Circles(b) How do we distinguish between real circles,

point circles and imaginary circles?

E2

( )2

+ － F＞ 0,D2

( )2

E2

( )2

+ － F = 0,D2

( )2

E2

( )2

+ － F＜ 0,D2

( )2

Determine the type of circle that each of the following

(2) x2+y2 － 16x － 8y+80 = 0

(1) x2+y2+8x+6y+15 = 0

(3) x2+y2+14x+4y+60 = 0

equations represents.

13. Locus and Equations of Circles13. Locus and Equations of Circles(b) How do we distinguish between real circles,

point circles and imaginary circles?

E.gE.g..

x2+y2+Dx+Ey+F = 0

E2

( )2+ － FD2

( )2

10= 0＞

∴ It is a real circle (1) x2+y2+8x+6y+15 = 0

13. Locus and Equations of Circles13. Locus and Equations of Circles(b) How do we distinguish between real circles,

point circles and imaginary circles?

= (4)2+(3)2 － 15

62

( )2

+ －1582

( )2

=

= ( － 8)2+( － 4)2 －80

x2+y2+Dx+Ey+F = 0

－82

( )2

+ －80－ 162

( )2

=

0=

∴ It is a point circle (2) x2+y2 － 16x － 8y+80 = 0

E2

( )2+ － FD2

( )2

13. Locus and Equations of Circles13. Locus and Equations of Circles(b) How do we distinguish between real circles,

point circles and imaginary circles?

= (7)2+(2)2 － 60

x2+y2+Dx+Ey+F = 0

－ 7= 0＜

∴ It is an imaginary circle (3) x2+y2+14x+4y+60 = 0

E2

( )2+ － FD2

( )2

13. Locus and Equations of Circles13. Locus and Equations of Circles(b) How do we distinguish between real circles,

point circles and imaginary circles?

42

( )2

+ －60142

( )2

=