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Length of the latus rectum of a hyperbola from its foci and eccentricity, a point or a directrix

This type of question, asked 4 times: 31 Jan 2023, 22 Jan 2025, 2 Apr 2025, 5 Apr 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    31 Jan 2023

    161
    Let HH be the hyperbola, whose foci are (1±2,0)(1 \pm\sqrt{2},0) and eccentricity is 2\sqrt{2}. Then the length of its latus rectum is

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  2. Question 2

    22 Jan 2025

    70
    Let the foci of a hyperbola be (1,14)(1,14) and (1,−12)(1, - 12). If it passes through the point (1,6)(1,6), then the length of its latus-rectum is :

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  3. Question 3

    2 Apr 2025

    52
    Let one focus of the hyperbola H:x2a2−y2b2=1H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1 be at (10,0)(\sqrt{10},0) and the corresponding directrix be  x=910\ ^{x=\frac{9}{\sqrt{10}}}. If e and  l\ ^{l} respectively are the eccentricity and the length of the latus rectum of HH, then 9(e2+l)9\left( e^{2}+l \right) is equal to:

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  4. Question 4

    5 Apr 2026 S2

    63
    Let the eccentricity e of a hyperbola satisfy the equation 6e2−11e+3=06e^{2}- 11e + 3 = 0. If the foci of the hyperbola are (3,5)(3,−4)(3,5)(3, - 4), then the length of its latus rectum is:

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