Question

If $ \theta $ is the angle subtended by a latus rectum at the center of the hyperbola having eccentricity $$ \frac{2}{\sqrt{7} - \sqrt{3}}, $$ then find $ \sin \theta $.

• A. \( \frac{1}{2} \tan \frac{\theta}{2} \)
• B. \( 2 \cos \frac{\theta}{2} \)
• C. \( \frac{1}{\sin \frac{\theta}{2} + \cos \frac{\theta}{2}} \)
• D. \( 1 - \cos \frac{\theta}{2} \)

Hint:

Relate eccentricity and latus rectum properties to angles using conic formulas.

Answer 1

The Correct Option is A

Use properties of hyperbola and latus rectum. Use formulas connecting eccentricity \( e \), latus rectum, and angle subtended. Derive expression for \( \sin \theta \).