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Tangent and normal to a hyperbola at a given point

This type of question, asked 4 times: 2021 Paper 10, 2021 Paper 19, 25 Jun 2022, 26 Jun 2022. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 10

    66
    Consider a hyperbola H:x2−2y2=4H:x^{2}- 2y^{2}= 4. Let the tangent at a point P(4,6)P(4,\sqrt{6}) meet the xx-axis at QQ and latus rectum at R(x1,y1),x1>0R\left( x_{1},y_{1} \right),x_{1}> 0. If FF is a focus of HH which is nearer to the point PP, then the area of △QFR\bigtriangleup QFR is equal to

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  2. Question 2

    2021 Paper 19

    65
    The point P(−26,3)P( - 2\sqrt{6},\sqrt{3}) lies on the hyperbola x2a2−y2b2=1\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1 having eccentricity 52\frac{\sqrt{5}}{2}. If the tangent and normal at PP to the hyperbola intersect its conjugate axis at the point QQ and RR respectively, then QRQR is equal to:

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  3. Question 3

    25 Jun 2022

    179
    Let the eccentricity of the hyperbola x2a2−y2b2=1\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1 be 54\frac{5}{4}. If the equation of the normal at the point (85,125)\left( \frac{8}{\sqrt{5}},\frac{12}{5} \right) on the hyperbola is 85x+βy=λ8\sqrt{5}x+\beta y=\lambda, then λ−β\lambda-\beta is equal to

  4. Question 4

    26 Jun 2022

    163
    The normal to the hyperbola x2a2−y29=1\frac{x^{2}}{a^{2}}-\frac{y^{2}}{9}= 1 at the point (8,33)(8,3\sqrt{3}) on it passes through the point :

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