Solve s=tsqrt{a^2+t^2}

$$s = t \sqrt { a ^ { 2 } + t ^ { 2 } }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\sqrt{\frac{s^{2}-t^{4}}{t^{2}}}\text{; }a=\sqrt{\frac{s^{2}-t^{4}}{t^{2}}}\text{, }&t\neq 0\text{ and }\left(|arg(\sqrt{\frac{s^{2}}{t^{2}}}t)-arg(s)|<\pi \text{ or }s=0\right)\\a\in \mathrm{C}\text{, }&s=0\text{ and }t=0\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=\frac{\sqrt{s^{2}-t^{4}}}{|t|}\text{; }a=-\frac{\sqrt{s^{2}-t^{4}}}{|t|}\text{, }&\left(s\geq t^{2}\text{ and }t>0\right)\text{ or }\left(s\leq -t^{2}\text{ and }t<0\right)\\a\in \mathrm{R}\text{, }&s=0\text{ and }t=0\end{matrix}\right.$$

Solve for s

$$s=t\sqrt{t^{2}+a^{2}}$$