Show that the pedal equation of the curve $x=a(2\cos ⁡t-\cos⁡2t),\ y=a(2\sin ⁡t-\sin⁡2t)$ is $9(r^2-a^2 )=8p^2$

Question

Show that the pedal equation of the curve $x=a(2\cos ⁡t-\cos⁡2t),\ y=a(2\sin ⁡t-\sin⁡2t)$ is $9(r^2-a^2 )=8p^2$.

I have found $$\frac{\mathrm dy}{\mathrm dx} = \frac{\cos θ-\cos2θ}{\sin2θ-\sin θ}.$$

Then I found the equation of tangent and applied the distance formula, but the equation gets complicated. Please, I really appreciate the help. Thanks in anticipation.

Answer 1

This is the required pedal equation. In this question, $x^2+y^2$ and $1+{(\frac{dy}{dx})}^2$ can be easily reduced to simpler terms, so one should try to proceed further even if equation starts looking complicated.