The minimum area of a triangle formed by any | vedclass

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(D) For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,the equation of the tangent at point $(a \cos 	heta, b \sin 	heta)$ is $\frac{x}{a} \cos

Vedclass · Accepted Answer

$36$

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(D) For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,the equation of the tangent at point $(a \cos 	heta, b \sin 	heta)$ is $\frac{x}{a} \cos 	heta + \frac{y}{b} \sin 	heta = 1$.The intercepts on the coordinate axes are $x = \frac{a}{\cos 	heta}$ and $y = \frac{b}{\sin 	heta}$.The area of the triangle formed by the tangent and the axes is $\Delta = \frac{1}{2} |x \cdot y| = \frac{1}{2} \left| \frac{ab}{\sin 	heta \cos 	heta} ight| = \left| \frac{ab}{\sin 2	heta} ight|$.Since the minimum value of $|\sin 2	heta|$ is $1$,the minimum area is $ab$.Here,$a^2 = 16 \implies a = 4$ and $b^2 = 81 \implies b = 9$.Therefore,the minimum area is $4 	imes 9 = 36$.

The minimum area of a triangle formed by any tangent to the ellipse $\frac{x^2}{16} + \frac{y^2}{81} = 1$ and the coordinate axes is

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