Find the polar of the point (a, - b) with respect to the circle $x^{2} + y^{2} + 2 a x - 2 b y + a^{2} - b^{2} = 0$

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Solution

The polar of the general equation of circle w.r.t point $(x_{1}, y_{1})$ is given as
$x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0$
Here for the circle $x^{2} + y^{2} + 2 a x - 2 b y + a^{2} - b^{2} = 0$ ,
$g = a, f = - b, c = a^{2} - b^{2}$
Hence equation of polar is
$x (a) + y (- b) + a (x + a) - b (y - b) + a^{2} - b^{2} = 0$
$a x - b y + a^{2} = 0$

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