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Hyperbola from two given facts, then its tangent of a given slope

The same question, asked 2 times: 28 Jun 2022*, 13 Apr 2023. Every wording below, oldest first.

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  1. Question 1

    28 Jun 2022

    73
    Let the eccentricity of the hyperbola H:x2a2−y2 b2=1H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{{\text{ }b}^{2}}= 1 be 52\sqrt{\frac{5}{2}} and length of its latus rectum be 626\sqrt{2}, If y=2x+cy = 2x + c is a tangent to the hyperbola HH, then the value of c2c^{2} is equal to

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  2. Question 2

    13 Apr 2023

    178
    The foci of a hyperbola are (±2,0)( \pm 2,0) and its eccentricity is 32\frac{3}{2}. A tangent, perpendicular to the line 2x+3y=62x + 3y = 6, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the xx - and yy-axes are aa and bb respectively, then ∣6a∣+∣5b∣|6a| + |5b| is equal to