question_answer

Let P be a variable point on the ellipse \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{16}=1\] with foci at \[{{S}_{1}}\] and \[{{S}_{2}}\]. If A is the area of triangle \[P{{S}_{1}}\,{{S}_{2}}\], then the maximum value of A is

A)  12                               

B)  24

C)  36                               

D)  48

Correct Answer: A

Solution :

\[{{e}^{2}}=1-\frac{16}{25}\,=\frac{9}{25}\]\[\Rightarrow \,e=\frac{3}{5}\] Area \[=\frac{1}{2}({{S}_{1}}{{S}_{2}})\,(altitude)\] \[=\frac{1}{2}(6)\,(altitude)\,=3((0,4)\And (0,0))=12\]