Area of the triangle formed by a tangent to the hyperbola xy=75 and its asymptotes

$$\begin{gathered} \mathrm{Given} \mathrm{hyperbola} \mathrm{xy}=75 \\ \mathrm{where} c^2=75 \Rightarrow c=5\sqrt{3} \\ \mathrm{Equation} \mathrm{of} \mathrm{tangent} \mathrm{at} \mathrm{P}\left(\mathrm{t}\right)=\left(ct, \frac{\mathrm{c}}{\mathrm{t}}\right) \mathrm{is} \\ \frac{x}{\mathrm{t}}+t\mathrm{y}=2c \\ \Rightarrow \frac{x}{\mathrm{t}}+t\mathrm{y}=10\sqrt{3} \\ \Rightarrow \frac{\mathrm{x}}{{\displaystyle 10\sqrt{3}t}}+\frac{\mathrm{y}}{{\displaystyle \frac{10\sqrt{3}}{ \mathrm{t}}}}=1 \\ Since \mathrm{equations} \mathrm{of} \mathrm{asymptotes} \mathrm{are} \mathrm{coordinate} \mathrm{axes} \\ \therefore \mathrm{Required} \mathrm{Area}=\frac{1}{2}\left|10\sqrt{3}t\times \frac{10\sqrt{3}}{ \mathrm{t}}\right| \\ =150 \end{gathered}$$