Question

If the length of the latusrectum of the ellipse $x^2{\tan }^2\alpha +y^2{\sec }^2\alpha =1$ is $\frac{1}{2}$,then $\alpha$ =

• A.
• B.
• C.
• D. none

Answer 1

The correct option is A

Given ellipse is $\frac{x^2}{{\cot }^2\alpha }+\frac{y^2}{{\cos }^2\alpha }=1$
Latus rectum is $\frac{2b^2}{a}=\frac{2{\cos }^2\alpha }{\cot \alpha }=\frac{1}{2}\Rightarrow 4{\cos }^2\alpha =\cot \alpha \Rightarrow 4\cos \alpha \sin \alpha =1\Rightarrow \sin 2\alpha =\frac{1}{2}\Rightarrow \alpha =\frac{\pi }{12}$