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  • If be a variable point on the ellipse <img alt="\mathrm{\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\

If \mathrm{P} be a variable point on the ellipse \mathrm{\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1} with foci \mathrm{\mathrm{F}_{1}} and \mathrm{\mathrm{F}_{2}}. If \mathrm{\mathrm{A}} is the area of the triangle \mathrm{\mathrm{PF}_{1} \mathrm{~F}_{2}} , then the maximum value of \mathrm{\mathrm{A}} is

Option: 1

2 abe


Option: 2

 abe


Option: 3

3 abe


Option: 4

\mathrm{a}^{2} \mathrm{~b}^{2} \mathrm{e}^{2}


Answers (1)

best_answer

Let \mathrm{P(a \cos \theta, b \sin \theta)} be a variable point on the ellipse \mathrm{\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1}

\mathrm{\mathrm{A}=\text{Area of}\, \mathrm{PF}_{1} \mathrm{~F}_{2} \Rightarrow \mathrm{A}=\frac{1}{2}( Base \times Height )}
\mathrm{\Rightarrow \mathrm{A}=\frac{1}{2}\left(\mathrm{~F}_{1} \mathrm{~F}_{2} \times \mathrm{PM}\right)}(M Is the foot of perpendicular from \mathrm{P} on the major axis)
\mathrm{\Rightarrow \mathrm{A}=\frac{1}{2}(2 \mathrm{ae} \times \mathrm{b} \sin \theta)}
\mathrm{\Rightarrow \mathrm{A}=\operatorname{abe} \sin \theta \mathrm{A}} is maximum when \mathrm{\theta=\pi / 2} and the maximum value of \mathrm{A} is abe.

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