Exam

JEE Main

Year

2019

Date

11 January

Shift

Morning Shift

Subject

Mathematics

Chapter

Ellipse

JEE Main 2019 · 11 January · Morning Shift · Mathematics · Ellipse

If tangents are drawn to the ellipse x^ 2 +2 y^ 2 =2 at all points...

If tangents are drawn to the ellipse $x^{2}+2 y^{2}=2$ at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve :

Step-by-step solution

Given the equation of ellipse, $\frac{x^{2}}{(\sqrt{2})^{2}}+y^{2}=1$
$\frac{\sqrt{2} \cos \theta x}{2}+y \sin \theta=1$ $P\left(\frac{\sqrt{2}}{\cos \theta}, 0\right)$ and $Q\left(0, \frac{1}{\sin \theta}\right)$ Let mid point be $(h, k)$ $\Rightarrow h=\frac{1}{\sqrt{2} \cos \theta}, k=\frac{1}{2 \sin \theta}$ As $\cos ^{2} \theta+\sin ^{2} \theta=1$ $\therefore \quad \frac{1}{2 h^{2}}+\frac{1}{4 k^{2}}=1$ Locus is $\frac{1}{2 x^{2}}+\frac{1}{4 y^{2}}=1$