Axes are co-ordinate axes, S and S¹ are Foci, B and B¹ are the ends of minor axis $\angle {\mathrm{SBS}}^{|}={\sin }^{-1}\left(\frac{4}{5}\right).$ If area of ${\mathrm{SBS}}^1{\mathrm{B}}^1$ is 20 sq. units then equation  of the ellipse is 

 

$$\begin{gathered} \mathrm{sin\theta }=\frac{4}{5}\Rightarrow \mathrm{cos\theta }=\frac{3}{5} \\ {\left(2\mathrm{ae}\right)}^2={\mathrm{a}}^2+{\mathrm{a}}^2-2{\mathrm{a}}^2\left(\frac{3}{5}\right) \\ 4{\mathrm{e}}^2=2-\frac{6}{5}=\frac{4}{5} \\ \mathrm{e}=\frac{1}{\sqrt{5}} \end{gathered}$$

area of $\mathrm{S} \mathrm{B} {\mathrm{S}}^1 {\mathrm{B}}^1$ is

$$\begin{gathered} 2(\frac{1}{2}\times 2ae\times b)=20 \\ \mathrm{abe}=10 \\ {\mathrm{a}}^2{\mathrm{b}}^2{\mathrm{e}}^2=100 \\ {\mathrm{a}}^2 {\mathrm{a}}^2(1-\frac{1}{5})\left(\frac{1}{5}\right)=100 \\ {\mathrm{a}}^4\left(1-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\right)=500 \\ {\mathrm{a}}^4=25\times 25 \\ \mathrm{a}=5 \\ {\mathrm{b}}^2={\mathrm{a}}^2\left(1-\frac{1}{5}\right) \\ =25\left(\frac{4}{5}\right)=20 \\ \therefore \mathrm{eq}uation of ellipse is \frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{20}=1 \end{gathered}$$