• # question_answer Let P be a variable point on the ellipse $\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$ with foci ${{F}_{1}}$ and ${{F}_{2}}$. If A is the area of the triangle $P{{F}_{1}}{{F}_{2}}$, then maximum value of A is [IIT 1994; Kerala (Engg.) 2005] A)            ab    B)            abe C)            $\frac{e}{ab}$                          D)            $\frac{ab}{e}$

$b\sqrt{{{a}^{2}}-{{b}^{2}}}$if $a>b;a\sqrt{{{b}^{2}}-{{a}^{2}}}$if $P$ Area of $P{{F}_{1}}{{F}_{2}}=\frac{1}{2}({{F}_{1}}{{F}_{2}})\times PL$                     $=\frac{1}{2}(2ac)\times y=ae.\frac{b}{a}\sqrt{{{a}^{2}}-{{x}^{2}}}$                   $A=eb\sqrt{{{a}^{2}}-{{x}^{2}}}$, which is maximum when$x=0$.                    Thus the maximum value of A is abe.