Mathematics | VITEEE 2014 | Discussion Page

The minimum area of the triangle formed by any tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with the coordinate axes is

$a^{2}+b^{2}$

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

C) ab

$\frac{(a-b)^{2}}{2}$

Option C

Equation of tangent at $(a \cos \theta,b \sin \theta)$ to the ellipse is

$\frac{x}{a} \cos \theta+\frac{y}{b} \sin \theta=1$

Coordinates of P and Q are

$\left(\frac{a}{\cos \theta},0\right)$ and $\left(0,\frac{b}{\sin \theta}\right)$ , respectively

Area of $\triangle OPQ$ = $\frac{1}{2}|\frac{a}{\cos \theta} \times \frac{b}{\sin \theta}|=\frac{ab}{|\sin 2 \theta|}$

$\therefore$ Minimum area =ab

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