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Focal distances of a point on a hyperbola

This type of question, asked 4 times: 8 Apr 2024, 3 Apr 2025, 4 Apr 2025, 7 Apr 2025. All 4 questions below, oldest first.

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  1. Question 1

    8 Apr 2024

    80
    Let H:−x2a2+y2 b2=1H:\frac{-x^{2}}{a^{2}}+\frac{y^{2}}{{\text{ }b}^{2}}= 1 be the hyperbola, whose eccentricity is 3\sqrt{3} and the length of the latus rectum is 434\sqrt{3}. Suppose the point (α,6),α>0(\alpha,6),\alpha> 0 lies on H. If β\beta is the product of the focal distances of the point (α,6)(\alpha,6), then α2+β\alpha^{2}+\beta is equal to:

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  2. Question 2

    3 Apr 2025

    72
    Let the product of the focal distances of the point P(4,23)P(4,2\sqrt{3}) on the hyperbola H:x2a2−y2 b2=1H:\frac{x^{2}}{a^{2}} - \frac{y^{2}}{{\text{ }b}^{2}} = 1 be 32.Let the length of the conjugate axis of H be p and the length of its latus rectum be qq. Then p2+q2p^{2} + q^{2} is equal to ......

  3. Question 3

    4 Apr 2025

    135
    Let the sum of the focal distances of the point P(4,3)P\left( 4,3 \right) on the hyperbola H:x2a2−y2 b2=H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{{\text{ }b}^{2}}= 11 be 8538\sqrt{\frac{5}{3}}. If for H , the length of the latus rectum is ll and the product of the focal distances of the point P is m , then 9l2+6 m9l^{2}+ 6\text{ }m is equal to :-

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  4. Question 4

    7 Apr 2025

    149
    Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2 a and 2 b , respectively, and one focus and the corresponding directrix of this hyperbola be (−5,0)( - 5,0) and 5x+9=05x + 9 = 0, respectively. If the product of the focal distances of a point (α,25)(\alpha,2\sqrt{5}) on the hyperbola is p , then 4 p is equal to .