Question

Solve this:
P²(x²-a²)-2pxy+y²-b²=0

Answer 1

$$p^2(x^2-a^2)-2pxy+y^2-b^2=0$$

$$p^2x^2-p^2a^2-2pxy+y^2-b^2=0$$

$$(y^2-2pxy+p^2x^2)-(p^2a^2+b^2)=0$$

$$(y-px)^2=p^2a^2+b^2$$

$$y=px\pm\sqrt{p^2a^2+b^2}$$

Each of the component equations is in Clairaut's form.

Hence changing $p$ to the arbitrary constant $c$ in

$$p^2(x^2-a^2)-2pxy+y^2-b^2=0$$

the required solution is

$$c^2(x^2-a^2)-2cxy+y^2-b^2=0$$

Or

$$y=cx\pm\sqrt{c^2a^2+b^2}, c=const$$