Question

If normal at any point $P$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ meet the major and minor axes at $Q$ and R respectively so that $3 PQ =2 PR$, then the eccentricity of ellipse is equal to

• A. $\frac{1}{\sqrt{3}}$
• B. $\sqrt{\frac{2}{3}}$
• C. $\frac{\sqrt{3}}{2}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

Answer: (A)

$ P(a \cos \theta, b \sin \theta)$
Equation of normal ax $\sec \theta-$ by $\operatorname{cosec} \theta= a ^2 e ^2$
$\therefore Q \left(\frac{ ae ^2}{\sec \theta}, 0\right) \& R \left(0, \frac{ a ^2 e ^2}{ b \operatorname{cosec} \theta}\right)$
Now $PQ : QR =2: 1$
Using section formula
$ae ^2 \cos \theta=\frac{2(0)+1( a \cos \theta)}{3}$
Hence $e ^2=\frac{1}{3} \Rightarrow e =\frac{1}{\sqrt{3}}$