Let P be a variable point on ellipse frac{{{x^2}}}{{{a^2}}} + frac{{{y^2}}}{{{b^2}}} = 1 with foci {

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10-2. Parabola, Ellipse, Hyperbola

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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

A

$ab$

B

$abe$

C

$\frac{e}{{ab}}$

D

$\frac{{ab}}{e}$

(IIT-1994)

Solution

(b) $b\sqrt {{a^2} – {b^2}} $ if $a > b;$

$a\sqrt {{b^2} – {a^2}} $ if $b>a$

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 $abe.$

Standard 11

Mathematics

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