Question

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

• A. $\frac{\pi }{4}or\frac{\pi }{3}$
• B. $\frac{\pi }{3}or\frac{3\pi }{5}$
• C. $\frac{\pi }{4}or\frac{3\pi }{4}$
• D. none of these.

Answer 1

Answer: (C)

Let the eccentric angle of the point be $\theta$ , then its coordinates are $(\sqrt{6}cos\theta ,\sqrt{2}sin\theta )$ Centre of a given ellipse is $(0,0)$
According to the equation,
$\sqrt{{\left(\sqrt{6}cos\theta -0\right)}^2+{\left(\sqrt{2}sin\theta -0\right)}^2}=2$
$\Rightarrow 6co{s}^2\theta +2si{n}^2\theta =4$
$\Rightarrow 6co{s}^2\theta +2\left(1-co{s}^2\theta \right)=4$
$\Rightarrow 4co{s}^2\theta =2$
$\Rightarrow cos\theta =\pm \frac{1}{\sqrt{2}}$
$\Rightarrow \theta =\frac{\pi }{4}$ or $\frac{3\pi }{4}$