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Condition for a line to touch a hyperbola

This type of question, asked 5 times: 14 Jun 2022, 28 Jun 2022, 25 Jul 2022, 13 Apr 2023, 22 Jan 2026 S1. All 5 questions below, oldest first.

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  1. Question 1

    14 Jun 2022

    75
    Let λx−2y=μ\lambda x- 2y=\mu be a tangent to the hyperbola a2x2−y2=b2a^{2}x^{2}-y^{2}=b^{2}. Then (λa)2−(μb)2\left( \frac{\lambda}{a} \right)^{2}-\left( \frac{\mu}{b} \right)^{2} is equal to:

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  2. Question 2

    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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  3. Question 3

    25 Jul 2022

    88
    Let the equation of two diameters of a circle x2+y2x^{2}+y^{2} −2x+2fy+1=0- 2x + 2fy + 1 = 0 be 2px−y=12px - y = 1 and 2x+py=4p2x + py = 4p. Then the slope m∈(0,∞)m \in (0,\infty) of the tangent to the hyperbola 3x2−y2=33x^{2}-y^{2}= 3 passing through the centre of the circle is equal to

  4. Question 4

    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

  5. Question 5

    22 Jan 2026 S1

    60
    If the line αx+2y=1\alpha x + 2y = 1, where α∈R\alpha \in R, does not meet the hyperbola x2−9y2=9x^{2}- 9y^{2}= 9, then a possible value of α\alpha is :

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