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Latus rectum of a hyperbola subtending a given angle at a point

This type of question, asked 2 times: 30 Jan 2024, 7 Apr 2025. All 2 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    30 Jan 2024

    83
    Let the latus rectum of the hyperbola x29−y2b2=1\frac{x^{2}}{9}-\frac{y^{2}}{b^{2}}= 1 subtend an angle of π3\frac{\pi}{3} at the centre of the hyperbola. If b2b^{2} is equal to l m(1+n)\frac{l}{\text{ }m}(1 +\sqrt{n}), where ll and mm are co-prime numbers, then l2+m2+n2l^{2}+m^{2}+n^{2} is equal to

  2. Question 2

    7 Apr 2025

    73
    Consider the hyperbola x2a2−y2b2=1\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 having one of its focus at P(−3,0)P( - 3,0). If the latus rectum through its other focus subtends a right angle at PP and a2b2=α2−β,α,β∈Na^{2}b^{2} = \alpha\sqrt{2} - \beta,\alpha,\beta \in \mathbb{N}, then α+β\alpha + \beta is equal to