Question

At some point $P$ on the ellipse, the segment $S{S}^1$ subtends a right angle, then its eccentricity is

• A. $e=\frac{\sqrt{2}}{2}$
• B. $e<\frac{1}{\sqrt{2}}$
• C. $e>\frac{1}{\sqrt{2}}$
• D. $\frac{\sqrt{3}}{2}$

Answer 1

The correct option is C $e>\frac{1}{\sqrt{2}}$

$m_1=\left(\frac{b\sin \theta -0}{a\cos \theta -l}\right)$
$m_2=\frac{(b\sin \theta -0)}{a\cos \theta +al}$
$m_1{m}_2=-1$
$\frac{b^2{\sin }^2\theta }{a^2{\cos }^2\theta -a^2{l}^2}=-1$
$a^2(1-l^2){\sin }^2\theta +a^2{\cos }^2\theta -a^2{l}^2=0$

$1=l^2(1+{\sin }^2\theta )$
$l^2=\frac{1}{{\sin }^2\theta +1}$
$l>\frac{1}{\sqrt{{\sin }^2\theta +1}}$
$e>\frac{1}{\sqrt{2}}(\because ma{x}^n$ value of ${\sin }^2\theta +1=2)$