Question

The eccentricity of an ellipse whose pair of a conjugate diameters 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 $\frac{1}{\sqrt{3}}$

For conjugate diameter $m_1\times m_2=\frac{-b^2}{a^2}$

where $m_1$ and $m_2$ are slopes of the respective diameters
$\Rightarrow 1\times \left(\frac{-2}{3}\right)=\frac{{-b}^2}{a^2}\Rightarrow b^2=\frac{2}{3}{a}^2$
Now $b^2=a^2(1-e^2)\Rightarrow \frac{2}{3}{a}^2=a^2(1-e^2)$
$\Rightarrow \frac{2}{3}=1-e^2\Rightarrow e^2=\frac{1}{3}\Rightarrow e=\frac{1}{\sqrt{3}}$