Solve a^{2}b^{2}-x^2b^2-y^2a^2=0

$$a ^ { 2 } b ^ { 2 } - x ^ { 2 } b ^ { 2 } - y ^ { 2 } a ^ { 2 } = 0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\left(b^{2}-y^{2}\right)^{-\frac{1}{2}}bx\text{; }a=\left(b^{2}-y^{2}\right)^{-\frac{1}{2}}bx\text{, }&b\neq -y\text{ and }b\neq y\\a\in \mathrm{C}\text{, }&\left(x=0\text{ and }b=y\right)\text{ or }\left(x=0\text{ and }b=-y\right)\text{ or }\left(b=0\text{ and }y=0\right)\end{matrix}\right.$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\left(a^{2}-x^{2}\right)^{-\frac{1}{2}}ay\text{; }b=\left(a^{2}-x^{2}\right)^{-\frac{1}{2}}ay\text{, }&a\neq -x\text{ and }a\neq x\\b\in \mathrm{C}\text{, }&\left(y=0\text{ and }a=x\right)\text{ or }\left(y=0\text{ and }a=-x\right)\text{ or }\left(a=0\text{ and }x=0\right)\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=\frac{bx}{\sqrt{b^{2}-y^{2}}}\text{; }a=-\frac{bx}{\sqrt{b^{2}-y^{2}}}\text{, }&|b|>|y|\\a=0\text{, }&\left(x=0\text{ and }|b|\neq |y|\right)\text{ or }\left(b=0\text{ and }y\neq 0\right)\\a\in \mathrm{R}\text{, }&\left(b=0\text{ and }y=0\right)\text{ or }\left(x=0\text{ and }|b|=|y|\right)\end{matrix}\right.$$

Solve for b

$$\left\{\begin{matrix}b=\frac{ay}{\sqrt{a^{2}-x^{2}}}\text{; }b=-\frac{ay}{\sqrt{a^{2}-x^{2}}}\text{, }&|a|>|x|\\b=0\text{, }&\left(y=0\text{ and }|a|\neq |x|\right)\text{ or }\left(a=0\text{ and }x\neq 0\right)\\b\in \mathrm{R}\text{, }&\left(a=0\text{ and }x=0\right)\text{ or }\left(y=0\text{ and }|a|=|x|\right)\end{matrix}\right.$$