Solve frac{y_{1}+1/2}{x_{1}-t}=x_{1}

$$\frac { y _ { 1 } + \frac { 1 } { 2 } } { x _ { 1 } - t } = x _ { 1 }$$

Solve for t

$$\left\{\begin{matrix}t=x_{1}-\frac{y_{1}}{x_{1}}-\frac{1}{2x_{1}}\text{, }&y_{1}\neq -\frac{1}{2}\text{ and }x_{1}\neq 0\\t\neq 0\text{, }&x_{1}=0\text{ and }y_{1}=-\frac{1}{2}\end{matrix}\right.$$

Solve for x_1

$$\left\{\begin{matrix}x_{1}=\frac{\sqrt{4y_{1}+t^{2}+2}+t}{2}\text{, }&\left(y_{1}\neq -\frac{1}{2}\text{ or }t<0\right)\text{ and }y_{1}\geq -\frac{t^{2}}{4}-\frac{1}{2}\\x_{1}=\frac{-\sqrt{4y_{1}+t^{2}+2}+t}{2}\text{, }&\left(y_{1}\neq -\frac{1}{2}\text{ or }t>0\right)\text{ and }y_{1}\geq -\frac{t^{2}}{4}-\frac{1}{2}\end{matrix}\right.$$