Pedal Equations?

A relation between the distance $r$ of any point on a given curve from the origin and the length of the perpendicular $p$ from the origin to the tangent at that point is called Pedal Equation of the curve.

Consider a curve represented by the equation:
$c^{2}(x^{2} + y^{2}) = x^{2}y^{2}$
where $c$ is some constant.
If the Pedal Equation of this curve can be written in the form:
$\frac{\alpha}{p^{\beta}} + \frac{\gamma}{r^{\beta}} = \frac{\alpha}{c^{\beta}}$
where $\alpha$ , $\beta$ , $\gamma$ are positive coprime integers. Find the value of $\alpha+\beta+\gamma$