Show that the envelope of circles at right angles to the radii vectors of $r^n=a^n\cos{n\theta}$ is
$$ r^{\frac{n}{1-n}}=a^{\frac{n}{1-n}}\cos{\frac{n\theta}{1-n}} $$
I took a log on both sides and differentiated the curve twice but still not getting what has been given to be proved. I think I am going in the wrong direction. Any help will be truly appreciated.