Question

If $\left(\sqrt{3}\right)bx+ay=2ab$ is tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ , then eccentric angle $\theta$ is

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{3}$

Answer 1

The correct option is B $\frac{\pi }{6}$

Equation of tangent at a point $(acos\theta ,bsin\theta )is\frac{x}{a}cos\theta +\frac{y}{b}sin\theta =1$
But, it is the same as $\frac{x}{a}\frac{\sqrt{3}}{2}+\frac{y}{b}.\frac{1}{2}=1$
$\therefore cos\theta =\frac{\sqrt{3}}{2},sin\theta =\frac{1}{2}\Rightarrow \theta =\frac{\pi }{6}$