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Q. 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 $k$ is equal to ___

JEE MainJEE Main 2021Conic Sections

Solution:

Tangent
$\frac{x \cos \theta}{b}+\frac{y \sin \theta}{2 a}=1$
image
So, area $(\Delta OAB )=\frac{1}{2} \times \frac{ b }{\cos \theta} \times \frac{2 a }{\sin \theta}$
$=\frac{2 a b}{\sin 2 \theta} \geq 2 a b$
$\Rightarrow k=2$