Solve {l}{y^{2}+xyp-x^{2}p^2=0}{(px-y)^2=(1+p^2)(x^2-y)}

$$\left. \begin{array} { l } { y ^ { 2 } + x y p - x ^ { 2 } p ^ { 2 } = 0 } \\ { ( p x - y ) ^ { 2 } = ( 1 + p ^ { 2 } ) ( x ^ { 2 } - y ) } \end{array} \right.$$

Solve for x, y

$$\left\{\begin{matrix}\\x=0\text{, }y=0\text{; }x=-\frac{\left(\sqrt{5}+1\right)p\left(p^{2}+1\right)\left(-2\sqrt{5}p^{2}+\sqrt{5}-3\right)}{4\left(5p^{4}+3\sqrt{5}p^{2}+1\right)}\text{, }y=-\frac{p^{2}\left(p^{2}+1\right)\left(-2\sqrt{5}p^{2}+\sqrt{5}-3\right)}{2\left(5p^{4}+3\sqrt{5}p^{2}+1\right)}\text{, }&\text{unconditionally}\\x=\frac{\left(\sqrt{5}-1\right)p\left(p^{2}+1\right)}{2\sqrt{5}p^{2}+\sqrt{5}-3}\text{, }y=-\frac{2p^{2}\left(p^{2}+1\right)}{2\sqrt{5}p^{2}+\sqrt{5}-3}\text{, }&|p|\neq \frac{\sqrt{30\sqrt{5}-50}}{10}\end{matrix}\right.$$