11. Suppose a right-angled triangle is inscribed in a circle of radius 100 . Let $\alpha$ and $\beta$ be its acute angles. If $\tan \alpha=4 \tan \beta$, find the area of the triangle.
11. Answer: 8000 . $\tan \alpha=4 \tan \beta=\frac{4}{\tan \alpha} \Rightarrow \tan \alpha=2$. The two legs are $\frac{200}{\sqrt{5}}$ and $\frac{400}{\sqrt{5}}$ respectively
$$
\text { Area }=\frac{1}{2} \times \frac{200}{\sqrt{5}} \times \frac{400}{\sqrt{5}}=8000 .
$$