First slide

Ellipse

Question

Let the line $y = m x$ and the ellipse $2 x^{2} + y^{2} = 1$ intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co–ordinate axes at $(- \frac{1}{3 \sqrt{2}}, 0)$ and $(0, β)$ , then $β$ is equal to:

Moderate

A

$\frac{2 \sqrt{2}}{3}$
B

$\frac{\sqrt{2}}{3}$
C

$\frac{2}{\sqrt{3}}$
D

$\frac{2}{3}$

Solution

Let P be $(x_{1}, y_{1})$
Equation of normal at P is $\frac{x}{2 x_{1}} - \frac{y}{y_{1}} = - \frac{1}{2}$
It passes through $(- \frac{1}{3 \sqrt{2}}, 0) \Rightarrow \frac{- 1}{6 \sqrt{2} x_{1}} = - \frac{1}{2} \Rightarrow x_{1} = \frac{1}{3 \sqrt{2}}$
So $y_{1} = \frac{2 \sqrt{2}}{3}$ (as P lies in 1st quadrant) ,
$β = \frac{y_{1}}{2} = \frac{\sqrt{2}}{3}$

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