Question

If normal at point $(6,2)$ to the ellipse passes through its nearest focus $(5,2),$ having centre at $(4,2),$ then its eccentricity is

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

Answer 1

The correct option is D $\frac{1}{2}$

Normal at point $P(6,2)$ to the ellipse passes through its focus $Q(5,2)$.
So, equation of normal is $y=2$ and $P$ must be one extremity of the major axis.

Now, $ae=QR=1$ (where $R$ is centre) and $a-ae=1.$


$\Rightarrow a=2$
$\Rightarrow b^2=a^2-a^2{e}^2=4-1=3$
$\therefore e=\sqrt{1-\frac{3}{4}}=\frac{1}{2}$