If minimum area of triangle formed by a tangent to ellipse frac{x^{2}}{b^{2}}+frac{y^{2}}{4 a^{2}}=1

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10-2. Parabola, Ellipse, Hyperbola

hard

If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{4 a^{2}}=1$ and the co-ordinate axis is $kab,$ then $\mathrm{k}$ is equal to ..... .

A

$1$

B

$3$

C

$2$

D

$7$

(JEE MAIN-2021)

Solution

Tangent

$\frac{x \cos \theta}{b}+\frac{y \sin \theta}{2 a}=1$

So, area $(\Delta \mathrm{OAB})=\frac{1}{2} \times \frac{\mathrm{b}}{\cos \theta} \times \frac{2 \mathrm{a}}{\sin \theta}$

$=\frac{2 \mathrm{ab}}{\sin 2 \theta} \geq 2 \mathrm{ab}$

$\Rightarrow \mathrm{k}=2$

Standard 11

Mathematics

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