Pedal Equations?

Calculus Level 4

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 \(\alpha+\beta+\gamma\)


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