Question

For some \( \theta\in\left(0,\frac{\pi}{2}\right) \), let the eccentricity and the length of the latus rectum of the hyperbola \[ x^2-y^2\sec^2\theta=8 \] be \( e_1 \) and \( l_1 \), respectively, and let the eccentricity and the length of the latus rectum of the ellipse \[ x^2\sec^2\theta+y^2=6 \] be \( e_2 \) and \( l_2 \), respectively. If \[ e_1^2=\frac{2}{e_2^2}\left(\sec^2\theta+1\right), \] then \[ \left(\frac{l_1l_2}{e_1^2e_2^2}\right)\tan^2\theta \] is equal to:

Hint:

Always convert conic equations to standard form before extracting parameters like eccentricity and latus rectum.

Answer 1

Correct Answer: 16