General Equation of a CircleThe equation of a circle with center (h, k) and

radius r is (x – h)2 + (y – k)2 = r2

Putting, - 2h = 2g, - 2k = 2f and (h2 + k2 – r2 ) = c, then

x2 + y2 + 2gx + 2fy + c = 0 This is the general equation of a circle

Now, simplifying this equationor, x2 – 2hx + h2 + y2 - 2ky + k2 = r2

or, x2 + y2 – 2hx – 2ky + h2 + k2 – r2 = 0

Some Examples

x2 + y2 - 10x – 2y + 10 = 0 X2 + y2 + 10x – 2y + 17 = 0

Center and radius of a circle.[In general formx2 + y2 + 2gx + 2fy + c = 0]• The general equation of a circle is • x2 + y2 + 2gx + 2fy + c = 0• or, x2 + 2gx + y2 + 2fy = - c• or, x2 + 2.x.g + g2 + y2 + 2.y.f2 + f2 = g2 + f2 – c• or, (x + g)2 + (y + f)2 = g2 + f2 – c

Comparing this equation with the equation (x – h)2 + (y – k)2 = r2, we get Hence, the center of the circle (h, k) = (- g, - f)

Find the center and radius of the circle represented by x2 + y2 + 4x – 6y + 4 = 0

• The equation of a circle is x2 + y2 + 4x – 6y + 4 = 0• Comparing with the equation x2 + y2 + 2gx + 2fy

+ c = 0, we get• 2g = 4 i.e. g = 2; 2f = - 6 i.e. f = - 3 and c = 4Therefore, center (h,k) = (-g,-f) = (-2, -(-3) = (-2,3)

=

= 3 units