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 minimum length of the line segment AB is _______ . [2023]

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Solution

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 minimum length of the line segment AB is _______ . [2023]

Ans.
(7)

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

Let $(4 \cos θ, 3 \sin θ)$ be the coordinates of the point at which the tangent is drawn.

Equation of tangent $\frac{x \cos θ}{4} + \frac{y \sin θ}{3} = 1$

Let $A = (4 \sec θ, 0)$

$B = (0, 3 c o s e c θ)$ $(Tangent intersects coordinate axes at A and B)$

$A B^{2} = 16 \sec^{2} θ + 9 c o s e c^{2} θ = 25 + 16 \tan^{2} θ + 9 \cot^{2} θ$

$A B^{2} \geq 25 + 2 \times \sqrt{16 \tan^{2} θ \times 9 \cot^{2} θ} = 49$

$A B \geq 7$

$∴$ $A B_{\min} = 7$

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