Question

Let axes of ellipse be coordinate axes, $S$ and $S^{\prime }$ be foci, $B$ and $B^{\prime }$ are the endpoints of the minor axis. If $\sin \left(\angle SB{S}^{\prime }\right)=\frac{4}{5}$ and area of $SB{S}^{\prime }{B}^{\prime }$ is $20$ sq. unit, then the equation of ellipse is

• A. $\frac{x^2}{20}+\frac{y^2}{16}=1$
• B. $\frac{x^2}{18}+\frac{y^2}{13}=1$
• C. $\frac{x^2}{25}+\frac{y^2}{16}=1$
• D. $\frac{x^2}{25}+\frac{y^2}{20}=1$

Answer 1

The correct option is D $\frac{x^2}{25}+\frac{y^2}{20}=1$

Given, area of $SB{S}^{\prime }{B}^{\prime }=20$
$\Rightarrow \frac{1}{2}\left(2ae\right)\times \left(2b\right)=20\Rightarrow aeb=10\Rightarrow a^2{e}^2=\frac{100}{b^2}\cdots \left(1\right)$
And
$\sin \theta =\frac{4}{5}\Rightarrow \tan \theta =\frac{4}{3}\Rightarrow \frac{4}{3}=\frac{2\tan \theta /2}{1-{\tan }^2\theta /2}\Rightarrow 2{\tan }^2\frac{\theta }{2}+3\tan \frac{\theta }{2}-2=0\Rightarrow (2\tan \frac{\theta }{2}-1)(\tan \frac{\theta }{2}+2)=0\Rightarrow \tan \frac{\theta }{2}=\frac{1}{2}(\because \tan \frac{\theta }{2}\ne -2,\theta \in (0,\pi \left)\right)\Rightarrow \frac{ae}{b}=\frac{1}{2}\cdots \left(2\right)$

From equations $\left(1\right)$ and $\left(2\right),$ we get
$b^2=20\Rightarrow a^2=25$

$\therefore$ Equation of ellipse will be
$\frac{x^2}{25}+\frac{y^2}{20}=1$