Question

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 $PF_{1}F_{2},$ then the maximum value of A is

• A. $2abe$
• B. $abe$
• C. $\frac{1}{2}abe$
• D. None of these

Answer 1

Answer: (B)

$abe$

Answer 2

Let $P(a\cos\theta,b\sin\theta)$ and $\mathbf{F}_{\mathbf{1}}\mathbf{(}\mathbf{-}\mathbf{ae,0),}\mathbf{F}_{\mathbf{2}}\mathbf{(ae,0)}$

$\mathbf{A =}$Area of $\Delta PF_{1}F_{2}$ $= \frac{1}{2}\left| \begin{matrix} a\cos\theta & b\sin\theta & 1 \ ae & 0 & 1 \

• ae & 0 & 1 \end{matrix} \right| = \frac{1}{2}|2aeb\sin\theta|$

$= aeb|\sin\theta|$

∴ A is maximum, when $|\sin\theta| = 1$.

Hence, maximum value of $A = abe$