Hyperbola and Ellipse Properties (Jee Mains)

Hyperbola

Hard

Hyperbola and Ellipse Properties

For some $\theta \in \left( {0,\frac{\pi }{2}} \right)$ let the eccentricity and the length of the latus rectum of the hyperbola x²–y²sec²θ = 8 be e₁ and ℓ₁, respectively, and let the eccentricity and the length of the latus rectum of the ellipse x²sec²θ + y² = 6 be e₂ and ℓ₂, respectively. If $e_1^2 = e_2^2({\sec ^2}\theta + 1)$ , then $\left( {\frac{{{\ell _1}{\ell _2}}}{{{e_1}{e_2}}}} \right){\tan ^2}\theta$ is equal to _____.

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Hyperbola

Hard

Hyperbola and Ellipse Properties

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