Solve y^2=(1-x^2/a^2)*b^2

$$y ^ { 2 } = ( 1 - \frac { x ^ { 2 } } { a ^ { 2 } } ) \cdot b ^ { 2 }$$

Solve for a (complex solution)

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

Solve for b (complex solution)

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

Solve for a

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

Solve for b

$$\left\{\begin{matrix}b=\sqrt{-\frac{1}{x^{2}-a^{2}}}ay\text{; }b=-\sqrt{-\frac{1}{x^{2}-a^{2}}}ay\text{, }&a\neq 0\text{ and }|x|<|a|\\b=0\text{, }&y=0\text{ and }a\neq 0\text{ and }|x|\neq |a|\\b\in \mathrm{R}\text{, }&y=0\text{ and }|x|=|a|\text{ and }a\neq 0\end{matrix}\right.$$