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Question

$T h e m i n i m u m a r e a o f t h e t r i a n g l e f o r m e d b y t h e \tan g e n t t o t h e e l l i p s e$ $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$
$a n d t h e c o o r d i n a t e a x e s i s$

Easy

A

ab
B

$\frac{a^{2} + b^{2}}{2}$
C

$\frac{{(a + b)}^{2}}{2}$
D

$\frac{a^{2} + a b + b^{2}}{3}$

Solution

$Equation of the tangent at P (θ) to the given ellipse is \frac{x}{a} \cos θ + \frac{y}{b} \sin θ = 1$
$∴Area of the triangle formed by it with coordinate axis = \frac{a b}{2 \sin θ \cos θ} = \frac{a b}{\sin 2θ} \geq a b$
$∴ m i n i m u n a r e a = a b$

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