Question
$$P$$ is a variable point on the ellipse $$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 2$$ whose foci are $${F_1}$$ and $${F_2}.$$ The maximum area $$\left( {{\text{in uni}}{{\text{t}}^2}} \right)$$ of the $$\Delta PFF'$$ is :
A.
$$2b\sqrt {{a^2} - {b^2}} $$
B.
$$\sqrt 2 b\sqrt {{a^2} - {b^2}} $$
C.
$$b\sqrt {{a^2} - {b^2}} $$
D.
$$2a\sqrt {{a^2} - {b^2}} $$
Answer :
$$2b\sqrt {{a^2} - {b^2}} $$
Solution :
Let $$P = \left( {\sqrt 2 a\cos \,\phi ,\,\sqrt 2 b\sin \,\phi } \right).\,{F_1}{\text{ and }}{F_2} = \left( { \pm \sqrt 2 ae,\,0} \right)$$
\[\begin{array}{l}
{\rm{The \,area\, of\, }}\Delta PFF' = \left| {\frac{1}{2}\left| \begin{array}{l}
\sqrt 2 a\cos \,\phi \,\,\,\,\,\sqrt 2 b\sin \,\phi \,\,\,\,\,\,\,\,\,\,\,\,1\\
\,\,\,\,\,\,\,\,\sqrt 2 ae\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\\
\,\,\, - \sqrt 2 ae\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1
\end{array} \right|} \right|\\
= \frac{1}{2}.\sqrt 2 b\sin \,\phi .\sqrt 2 ae\\
= 2abe\sin \,\phi
\end{array}\]
$$\therefore $$ maximum area $$ = 2abc = 2ab.\frac{{\sqrt {{a^2} + {b^2}} }}{a}$$