Question

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

Answer 1

Given curve,

So, general point on the ellipse is 

= (4cos ⁡θ, 3sin⁡ θ)

We know,

Equation of tangent to a given ellipse at its point (a cos θ, b sin⁡⁡ θ) is

\(\frac{x\,cos\,\theta}{a}+\frac{y\,sin\,\theta}{b}=1\)

∴ Here equation of tangent at point (4cos⁡ θ, 3sin⁡θ) is

\(\frac{x\,cos\,\theta}{4}+\frac{y\,sin\,\theta}{3}=1\)

When this tangent cut's x axis then y = 0.

∴ Point of intersection at x axis is A(4sec⁡ θ,0).

When this tangent cut's y axis then x = 0.

∴ Point of intersection at y axis is B(0,3cosec ⁡θ).

∴ Length of AB

∴ Minimum length of AB = 7.

Answer 2

Equation of tangent at point P(4cos\(\theta\), 3sin\(\theta\)) is