Hyperbola-locus of P equidist from asymptotes and x-axis?

A hyperbola of the form

$\displaystyle \frac{x^2}{\alpha}-\frac{y^2}{\beta}=1$

has asymptotes with equation $\displaystyle y^2=m^2 x^2$ and passes through the point (a,0). Find an equation of the hyperbola in terms of x, y, a and m.

Ans:$\displaystyle \frac{x^2}{a^2}-\frac{y^2}{a^2 m^2}=1$

A point P on this hyperbola is equidistant from one of its asymptoes and the x-axis. Prove that, for all values of m, P lies on the curve with equation

$\displaystyle (x^2-y^2)^2=4x^2 (x^2-a^2)$

Stuck here please help.