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 $\frac{1}{\sqrt{3}}$

In case of conjugated determine of ellipse

$m_1.m_2=\frac{-b^2}{a^2}$

So

$y=x\Rightarrow m_1=1$

$3y=-2x\Rightarrow m_2=\frac{-2}{3}$

$1\times \frac{-2}{3}=\frac{-b^2}{a^2}$

$\frac{b^2}{a^2}=\frac{2}{3}$

$e=\sqrt{1-\frac{b^2}{a^2}}$

$=\sqrt{1-\frac{2}{3}}$

$=\frac{1}{\sqrt{3}}$