Solve t-1/t-3-a^-2/t+3-x=1

$$\frac { t - 1 } { t - 3 } - \frac { a ^ { - 2 } } { t + 3 - x } = 1$$

Solve for a (complex solution)

$$a=-\frac{i\left(3+t-x\right)^{-\frac{1}{2}}\sqrt{6-2t}}{2}$$

$$a=\frac{i\left(3+t-x\right)^{-\frac{1}{2}}\sqrt{6-2t}}{2}\text{, }t\neq 3\text{ and }t\neq x-3$$

Solve for t (complex solution)

$$\left\{\begin{matrix}t=-\frac{3+6a^{2}-2xa^{2}}{2a^{2}-1}\text{, }&a\neq 0\text{ and }x\neq 6\text{ and }a\neq -\frac{\sqrt{2}}{2}\text{ and }a\neq \frac{\sqrt{2}}{2}\\t\in \mathrm{C}\setminus 3,x-3\text{, }&\left(a=\frac{\sqrt{2}}{2}\text{ and }x=6\right)\text{ or }\left(a=-\frac{\sqrt{2}}{2}\text{ and }x=6\right)\end{matrix}\right.$$

Solve for a

$$a=\frac{\sqrt{\frac{2\left(t-3\right)}{3+t-x}}}{2}$$

$$a=-\frac{\sqrt{\frac{2\left(t-3\right)}{3+t-x}}}{2}\text{, }\left(t>-\left(3-x\right)\text{ and }t>3\right)\text{ or }\left(t<-\left(3-x\right)\text{ and }t<3\right)$$

Solve for t

$$\left\{\begin{matrix}t=-\frac{3+6a^{2}-2xa^{2}}{2a^{2}-1}\text{, }&a\neq 0\text{ and }x\neq 6\text{ and }|a|\neq \frac{\sqrt{2}}{2}\\t\neq 3\text{, }&x=6\text{ and }|a|=\frac{\sqrt{2}}{2}\end{matrix}\right.$$