Question

The distance of a point on the ellipse $\frac{x^2}{6}+\frac{y^2}{2}=1$ from the center is $2$ unit. The eccentric angle of the point, is

• A. $\frac{\pi }{4}$
• B. $\frac{3\pi }{4}$
• C. $\frac{5\pi }{6}$
• D. $\frac{\pi }{6}$

Answer 1

Answer: (A), (B)

The correct options are
A $\frac{\pi }{4}$
B $\frac{3\pi }{4}$

Let $P(\sqrt{6}\cos \theta ,\sqrt{2}\sin \theta )$ be any point on the ellipse.
Let $O$ be the center.
$OP=2$
$\Rightarrow 6{\cos }^2\theta +2{\sin }^2\theta =4$
$\Rightarrow 3{\cos }^2\theta +{\sin }^2\theta =2$
$\Rightarrow 3{\cos }^2\theta +1-{\cos }^2\theta =2$
$\Rightarrow 2{\cos }^2\theta =1$
$\Rightarrow \cos \theta =\pm \frac{1}{\sqrt{2}}$
$\Rightarrow \theta =\frac{\pi }{4},\frac{3\pi }{4}$