Find the coordinates of the centres and the radii of the circles whose equation is:
$x^{2} + y^{2} = 2 g x - 2 f y$ .

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Solution

Given circle can be written as, $x^{2} + y^{2} - 2 g x + 2 f y = 0$
Now comparing with general equation of circle, $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
we get centre of the circle $= (- g, - f) = (g, - f)$
and radius $= \sqrt{g^{2} + f^{2} - c} = \sqrt{g^{2} + f^{2}}$ , as $c = 0$

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