Find the polar of the point $\left(-7, -9\right)$ with respect to the circle $x^2 + y^2 - 12x - 8y - 48 = 0$.
Given point: $\;\;$ $\left(h, k\right) = \left(-7, -9\right)$
Given circle: $\;\;$ $x^2 + y^2 - 12x - 8y - 48 = 0$
Comparing the equation of the given circle with the standard equation of circle $\;\;$ $x^2 + y^2 + 2gx + 2fy + c = 0$
gives $\;\;$ $g = -6, \; f = -4, \; c = -48$
The polar of $\;$ $\left(h, k\right)$ $\;$ with respect to the circle $\;$ $x^2 + y^2 + 2gx + 2fy + c = 0$ $\;$ is $\;$ $xh + yk + g \left(x + h\right) + f \left(y + k\right) + c = 0$
$\therefore \;$ The required polar is:
$-7x -9y - 6 \left(x - 7\right) - 4 \left(y - 9\right) - 48 = 0$
i.e. $\;$ $-7x -9y - 6x + 42 - 4y + 36 - 48 = 0$
i.e. $\;$ $-13x - 13y + 30 = 0$