Let a tangent be drawn to the ellipse $\frac{x^{2}}{27}+y^{2}=1$ at $(3 \sqrt{3} \cos \theta, \sin \theta)$ where $\theta \in\left(0, \frac{\pi}{2}\right)$. Then the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to :

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Ellipse

Let a tangent be drawn to the ellipse

\frac{x ^{2}}{27} + y^{2} = 1

(33 cos θ, sin θ)

where

θ \in (0, \frac{π}{2})

. Then the value of

θ

such that the sum of intercepts on axes made by this tangent is minimum is equal to :

Answer: (A)

The tangent to

\frac{x ^{2}}{27} + y^{2} = 1

(33 cos θ, sin θ)

\frac{x cos θ}{3 3} + y sin θ = 1

so it cuts the axes at

33 sec θ

and

cosec θ

. The sum of the intercepts is

S = 33 sec θ + cosec θ

, with

\frac{d S}{d θ} = \frac{3 3 sin θ}{cos ^{2} θ} - \frac{cos θ}{sin ^{2} θ}

This vanishes when

33 sin^{3} θ = cos^{3} θ

, that is

tan^{3} θ = 3^{- 3/2}

, giving

tan θ = \frac{1}{3}

. Since

S \to \infty

at both ends of

(0, \frac{π}{2})

, this single stationary point is the minimum, and it occurs at

θ = \frac{π}{6}

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Single correct option