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  4. Let a tangent to the curve 9 x2+16 y2=144 intersect the coordinate axes at the points A and B. Then, the mininum length of the line segment AB is

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Q. Let a tangent to the curve $9 x^2+16 y^2=144$ intersect the coordinate axes at the points $A$ and $B$. Then, the mininum length of the line segment $AB$ is

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Solution:

Equation of tangent at point $P (4 \cos \theta, 3 \sin \theta)$ is $\frac{ x \cos \theta}{4}+\frac{ y \sin \theta}{3}=1$
So $A$ is $(4 \sec \theta, 0)$ and point $B$ is $(0,3 \operatorname{cosec} \theta)$
Length $A B =\sqrt{16 \sec ^2 \theta+9 \operatorname{cosec}^2 \theta}$ $=\sqrt{25+16 \tan ^2 \theta+9 \cot ^2 \theta} \geq 7$