Question

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

• A. $\frac{x^{2}}{2} + \frac{y^{2}}{4} = 1 $
• B. $\frac{x^{2}}{4} + \frac{y^{2}}{2} = 1 $
• C. $\frac{1}{2x^{2}} + \frac{1}{4y^{2}} = 1 $
• D. $\frac{1}{4x^{2}} + \frac{1}{2y^{2}} = 1 $

Answer 1

Answer: (C)

Equation of general tangent on ellipse
$\frac{x}{a \sec\theta} + \frac{y}{b \cos ec\theta} = 1$
$ a=\sqrt{2} , b=1 $
$ \Rightarrow \frac{x}{\sqrt{2} \sec\theta} + \frac{y}{\cos ec\theta} = 1 $
Let the midpoint be $(h, k)$
$ h = \frac{\sqrt{2} \sec\theta}{2} \Rightarrow \cos\theta = \frac{1}{\sqrt{2}h} $
and $ k = \frac{\cos ec \theta}{2} \Rightarrow \sin \theta = \frac{1}{2k} $
$ \because \sin^{2} \theta + \cos^{2} \theta = 1 $
$ \Rightarrow \frac{1}{2h^{2} } + \frac{1}{ 4k^{2}} = 1 $
$ \Rightarrow \frac{1}{2x^{2}} + \frac{1}{4y^{2}} = 1 $