First slide

Tangents and normals

Question

Tangent is drawn to ellipse $\frac{x^{2}}{27} + y^{2} = 1 a t (3 \sqrt{3} \cos θ, \sin θ)$ $(where θ \in (0, π / 2))$ . Then the value of such that sum of intercepts on
axes made by this tangent is least is

Moderate

A

$π / 3$
B

$π / 6$
C

$π / 8$
D

$π / 4$

Solution

$Equation of tangent at (3 \sqrt{3} \cos θ, \sin θ) is$ $\frac{x \cos θ}{3 \sqrt{3}} + y \sin θ = 1$
Its intercept on coordinates axes are $3 \sqrt{3} / \cos θ a n d \cos e c θ$ If S denotes their sum, then
$\begin{array}{l} S = 3 \sqrt{3} \sec θ + cosec θ \\ \frac{d S}{d θ} = 3 \sqrt{3} \sec θ \tan θ - cosec θ \cot θ \\ For minimum value of S \cdot \frac{d S}{d θ} = 0 \\ \Rightarrow \frac{3 \sqrt{3} \sin θ}{\cos^{2} θ} - \frac{\cos θ}{\sin^{2} θ} = 0 \Rightarrow \tan^{3} θ = 3 \sqrt{3} \end{array}$
$\begin{array}{l} Since θ \in (0, π / 2), so \tan θ = \sqrt{3} \Rightarrow θ = π / 3 \\ Also \frac{d^{2} S}{d θ^{2}} = 3 \sqrt{3} \sec θ \tan^{2} θ + 3 \sqrt{3} \sec^{3} θ + {cosec}^{2} θ \\ \cot θ + {cosec}^{3} θ \\ {\Rightarrow \frac{d^{2} S}{d θ^{2}}|}_{θ - π / 3} > 0 . \underline{Thus S} is least when θ = π / 3 \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App