$xy =1,-1$

$\frac{t_{1}+t_{2}}{2} \cdot \frac{\frac{1}{t_{1}}-\frac{1}{t_{2}}}{2}=1$

$\Rightarrow t_{1}^{2}-t_{2}^{2}=4 t_{1} t_{2}$

$\frac{1}{t_{1}^{2}} \times\left(-\frac{1}{t_{2}^{2}}\right)=-1 \Rightarrow t_{1} t_{2}=1$

$\Rightarrow\left(t_{1} t_{2}\right)^{2}=1 \Rightarrow t_{1} t_{2}=1$

$t_{1}^{2}-t_{2}^{2}=4$

$\Rightarrow t_{1}^{2}+t_{2}^{2}=\sqrt{4^{2}+4}=2 \sqrt{5}$

$A B^{2}=\left(t_{1}-t_{2}\right)^{2}+\left(\frac{1}{t_{1}}+\frac{1}{t_{2}}\right)^{2}$

$\Rightarrow t_{1}^{2}=2+\sqrt{5} \Rightarrow \frac{1}{t_{1}^{2}}=\sqrt{5}-2$

$=2\left(t_{1}^{2}+\frac{1}{t_{1}^{2}}\right)=4 \sqrt{5} \Rightarrow$ Area $^{2}=80$