Find the polar of the point $\left(h, k\right)$ with respect to the circle $x^2 + y^2 - 2gx - 2fy + c = 0$.
Given point: $\;\;$ $\left(h, k\right)$
Given circle: $\;\;$ $x^2 + y^2 - 2gx - 2fy + c = 0$
Comparing the equation of the given circle with the standard equation of circle $\;\;$ $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$
gives $\;\;$ $g_1 = -g, \; f_1 = -f, \; c_1 = c$
The polar of $\;$ $\left(h, k\right)$ $\;$ with respect to the circle $\;$ $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ $\;$ is
$xh + yk + g_1 \left(x + h\right) + f_1 \left(y + k\right) + c_1 = 0$
$\therefore \;$ The required polar is:
$xh + yk -g \left(x + h\right) -f \left(y + k\right) + c = 0$
i.e. $\;$ $xh + yk -gx -gh -fy -fk + c = 0$
i.e. $\;$ $x \left(h - g\right) + y \left(k - f\right) -gh -fk + c = 0$