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If tangents are drawn to the ellipse

Question:

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 betwen the coordinate axes lie on the curve :

  1. $\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$

  2. $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$

  3. $\frac{1}{2 x^{2}}+\frac{1}{4 y^{2}}=1$

  4. $\frac{1}{4 x^{2}}+\frac{1}{2 y^{2}}=1$

Correct Option: , 3

Solution:

Equation of general tangent on ellipse

$\frac{\mathrm{x}}{\mathrm{a} \sec \theta}+\frac{\mathrm{y}}{\mathrm{bcosec} \theta}=1$

$a=\sqrt{2}, b=1$

$\Rightarrow \frac{x}{\sqrt{2} \sec \theta}+\frac{y}{\operatorname{cosec} \theta}=1$

Let the midpoint be $(h, k)$

$\mathrm{h}=\frac{\sqrt{2} \sec \theta}{2} \Rightarrow \cos \theta=\frac{1}{\sqrt{2} \mathrm{~h}}$

and $k=\frac{\operatorname{cosec} \theta}{2} \Rightarrow \sin \theta=\frac{1}{2 k}$

$\because \sin ^{2} \theta+\cos ^{2} \theta=1$

$\Rightarrow \frac{1}{2 \mathrm{~h}^{2}}+\frac{1}{4 \mathrm{k}^{2}}=1$

$\Rightarrow \frac{1}{2 \mathrm{x}^{2}}+\frac{1}{4 \mathrm{y}^{2}}=1$

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