Question

Find the polar of the point $(5,-\frac{1}{2})$ with respect to the circle $3x^2+3y^2-7x+8y-9=0.$

Answer 1

The equation of polar w.r.t to the point $(x_1,y_1)$ for a general equation of circle is

$x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c=0$

For the given circle $3x^2+3y^2-7x+8y-9=0$,

$g=\frac{-7}{6},f=\frac{4}{3},c=-3$

So, polar is

$x\left(5\right)+y\left(\frac{-1}{2}\right)-\frac{7}{6}(x+5)+\frac{4}{3}(y-\frac{1}{2})-3=0$

$23x+5y=57$