Question

The eccentricity of an ellipse whose pair of a conjugate diameter are $y=x$ and $3y=-2x$ is

• A. $\frac{2}{3}$
• B. $\frac{1}{3}$
• C. $\frac{1}{\sqrt{3}}$
• D. None of these

Answer 1

The correct option is C $\frac{1}{\sqrt{3}}$

Let the ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. since, $y=x$ and $3y=-2x$ is a pair of conjugate diameters.
Therefore, $m_1{m}_2=\frac{b^2}{a^2}$
$\Rightarrow \left(1\right)(-\frac{2}{3})=\frac{b^2}{a^2}$
$\Rightarrow 2a^2=3b^2$
$\Rightarrow 2a^2=3a^2(1-e^2)$
$\Rightarrow 2=3(1-e^2)$
$\Rightarrow e^2=\frac{1}{3}$
$\Rightarrow e=\frac{1}{\sqrt{3}}$