Solve x^{2}p^2+y^2q^2=z^2

$$x ^ { 2 } p ^ { 2 } + y ^ { 2 } q ^ { 2 } = z ^ { 2 }$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=-\frac{i\sqrt{\left(qy\right)^{2}-z^{2}}}{x}\text{; }p=\frac{i\sqrt{\left(qy\right)^{2}-z^{2}}}{x}\text{, }&x\neq 0\\p\in \mathrm{C}\text{, }&\left(y=\frac{z}{q}\text{ and }q\neq 0\text{ and }x=0\right)\text{ or }\left(y=-\frac{z}{q}\text{ and }q\neq 0\text{ and }x=0\right)\text{ or }\left(z=0\text{ and }q=0\text{ and }x=0\right)\end{matrix}\right.$$

Solve for q (complex solution)

$$\left\{\begin{matrix}q=-\frac{i\sqrt{\left(px\right)^{2}-z^{2}}}{y}\text{; }q=\frac{i\sqrt{\left(px\right)^{2}-z^{2}}}{y}\text{, }&y\neq 0\\q\in \mathrm{C}\text{, }&\left(x=\frac{z}{p}\text{ and }p\neq 0\text{ and }y=0\right)\text{ or }\left(x=-\frac{z}{p}\text{ and }p\neq 0\text{ and }y=0\right)\text{ or }\left(p=0\text{ and }z=0\text{ and }y=0\right)\end{matrix}\right.$$

Solve for p

$$\left\{\begin{matrix}p=\frac{\sqrt{z^{2}-\left(qy\right)^{2}}}{|x|}\text{; }p=-\frac{\sqrt{z^{2}-\left(qy\right)^{2}}}{|x|}\text{, }&x\neq 0\text{ and }|z|\geq |q||y|\\p\in \mathrm{R}\text{, }&\left(z=0\text{ and }q=0\text{ and }x=0\right)\text{ or }\left(q\neq 0\text{ and }x=0\text{ and }|y|=\frac{|z|}{|q|}\right)\end{matrix}\right.$$

Solve for q

$$\left\{\begin{matrix}q=\frac{\sqrt{z^{2}-\left(px\right)^{2}}}{|y|}\text{; }q=-\frac{\sqrt{z^{2}-\left(px\right)^{2}}}{|y|}\text{, }&\left(p=0\text{ or }|x|\leq \frac{|z|}{|p|}\right)\text{ and }y\neq 0\\q\in \mathrm{R}\text{, }&\left(p=0\text{ and }z=0\text{ and }y=0\right)\text{ or }\left(p\neq 0\text{ and }y=0\text{ and }|x|=\frac{|z|}{|p|}\right)\end{matrix}\right.$$