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 $3PQ=2PR$, 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 $\mathrm{cosec}\theta =a^2{e}^2$
$\therefore Q\left(\frac{a{e}^2}{\sec \theta },0\right)\&R\left(0,\frac{a^2{e}^2}{b\mathrm{cosec}\theta }\right)$
Now $PQ:QR=2:1$
Using section formula
$a{e}^2\cos \theta =\frac{2(0)+1(a\cos \theta )}{3}$
Hence $e^2=\frac{1}{3}\Rightarrow e=\frac{1}{\sqrt{3}}$