Question

If $bx+\sqrt{3}ay=2ab$ touches the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, then the eccentric angle of the point of contact is

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. None of these

Answer 1

Answer: (C)

$\frac{\pi }{3}$

Equation of tangent is $\frac{x}{a}.\frac{1}{2}+\frac{y}{b}.\frac{\sqrt{3}}{2}=1$ ....(1)

Also, equation of tangent at the point $(a\cos \theta ,b\sin \theta )$ is

$\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$ ...(2)

Since (1) and (2) are identical, we get

$\cos \theta =\frac{1}{2}$ and $\sin \theta =\frac{\sqrt{3}}{2}\Rightarrow \theta =\frac{\pi }{3}$