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Triangle formed by the centre and a double ordinate of a hyperbola

This type of question, asked 2 times: 23 Jan 2026 S2, 4 Apr 2026 S2. All 2 questions below, oldest first.

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  1. Question 1

    23 Jan 2026 S2

    59
    Let PQ be a chord of the hyperbola x24−y2 b2=1\frac{x^{2}}{4}-\frac{y^{2}}{{\text{ }b}^{2}}= 1, perpendicular to the x -axis such that OPQ is an equilateral triangle, O being the centre of the hyperbola. If the eccentricity of the hyperbola is 3\sqrt{3}, then the area of the triangle OPQ is :

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  2. Question 2

    4 Apr 2026 S2

    61
    Let H:x2a2−y2 b2=1H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{{\text{ }b}^{2}}= 1 be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is 83\frac{8}{3}. If the line x=αx =\alpha intersects the hyperbola H at the points A and B such that the area of the triangle AOB is 4154\sqrt{15}, where O is the origin, then α2\alpha^{2} equals

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