P is a variable point on the ellipse (x2/a2) + (y2/b2) = 1 with foci F1 and F2 . If A is the area of the triangle PF1F2. then the maximum value of A is

Question

P is a variable point on the ellipse (x2/a2) + (y2/b2) = 1 with foci F1 and F2 . If A is the area of the triangle PF1F2. then the maximum value of A is (A) (e/ab) (B) (ab/e) (C) aeb (D) (ab/a)

Tardigrade Admin · Accepted Answer

Let point P(a cos θ, b sin θ) on the ellipse (x2/a2)+(y2/b2)=1 ∴ Area of Δ P F1 F2=(1/2)(2 a e) b| sin θ|=a e b| sin θ|=A For maximum value of A, θ=(π/2) or (3 π/2) so A max =a e b .

Q. $P$ is 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 the maximum value of $A$ is

$\frac{e}{ab}$

$\frac{ab}{e}$

$a e b$

$\frac{ab}{a}$

Let point $P (a cos θ, b sin θ)$ on the ellipse
$\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$
$∴$ Area of $Δ P F_{1} F_{2} = \frac{1}{2} (2 a e) b ∣ sin θ ∣ = a e b ∣ sin θ ∣ = A$
For maximum value of
$A, θ = \frac{π}{2}$ or $\frac{3 π}{2}$
so $A_{m a x} = a e b .$

Q. P is a variable point on the ellipse a2x2​+b2y2​=1 with foci F1​ and F2​ . If A is the area of the triangle PF1​F2​. then the maximum value of A is

abe​

eab​

aeb

aab​