Mathematics - Area Under Curves Question with Solution | TestHub

The ellipse has eccentricity $\frac{4}{5}$, foci $(\pm 4,0)$, and $\left(4,-\frac{9}{5}\right)$ is an end point of the latus rectum. The circle has radius squared $4^2+{\left(\frac{9}{5}\right)}^2=\frac{481}{25}$, and the common chord $x=4$ subtends an angle $2{\tan }^{-1}\left(\frac{9/5}{4}\right)=2{\tan }^{-1}\left(\frac{9}{20}\right)$ at the centre. Required area = sector $-△POQ=\frac{1}{2}\cdot \frac{481}{25}\cdot 2{\tan }^{-1}\left(\frac{9}{20}\right)-\frac{1}{2}\cdot 4\cdot \frac{18}{5}=\frac{481}{25}{\tan }^{-1}\left(\frac{9}{20}\right)-\frac{36}{5}$.