Question: Tangent is drawn to ellipse $\frac{x^{2}}{27} + y^{2} = 1$are \(( 3 \sqrt { 3 } \cos \theta , \sin...

Question

Tangent is drawn to ellipse $\frac{x^{2}}{27} + y^{2} = 1$ are $( 3 \sqrt { 3 } \cos \theta , \sin \theta )$ $\left\lbrack where\theta\varepsilon\left( 0,\frac{\pi}{2} \right) \right\rbrack$ . Then the value of θ such that sum of intercepts on axes made by this tangent is minimum is

Answer 1

Solution

$\frac{x\cos\theta}{3\sqrt{3}} + y\sin\theta = 1$

Sum of intercepts = 3 $\sqrt{3}$ secθ + cosec θ = f(θ) (say).

⇒ f' (θ) = $\frac{3\sqrt{3}\sin^{3}\theta - \cos^{3}\theta}{\sin^{2}\theta\cos^{2}\theta}$

⇒ at θ = π/6, f(θ) is minimum.

Question: Tangent is drawn to ellipse \(\frac{x^{2}}{27} + y^{2} = 1\)are \(( 3 \sqrt { 3 } \cos \theta , \sin...

Solution