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 $k=\frac{\cos ec\theta }{2}\Rightarrow \sin \theta =\frac{1}{2k}$
$\because {\sin }^2\theta +{\cos }^2\theta =1$
$\Rightarrow \frac{1}{2h^2}+\frac{1}{4k^2}=1$
$\Rightarrow \frac{1}{2x^2}+\frac{1}{4y^2}=1$