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Common tangent to a circle and a parabola

This type of question, asked 6 times: 2021 Paper 4, 25 Jun 2022, 26 Jun 2022, 28 Jul 2022, 30 Jan 2023, 10 Apr 2023. All 6 questions below, oldest first.

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  1. Question 1

    2021 Paper 4

    82
    A line is a common tangent to the circle (x−3)2+y2=9(x- 3)^{2}+y^{2}= 9 and the parabola y2=4xy^{2}= 4x. If the two points of contact (a,b)(a,b) and (c,d)(c,d) are distinct and lie in the first quadrant, then 2(a+c)2(a+c) is equal to

  2. Question 2

    25 Jun 2022

    76
    If y=m1x+c1y =m_{1}x +c_{1} and y=m2x+c2, m1≠m2y =m_{2}x +c_{2},{\text{ }m}_{1}\neq m_{2} are two common tangents of circle x2+y2=2x^{2}+y^{2}= 2 and parabola y2=xy^{2}= x, then the value of 8∣ m1 m2∣8\left| {\text{ }m}_{1}{\text{ }m}_{2} \right| is equal to

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  3. Question 3

    26 Jun 2022

    83
    Let the common tangents to the curves 4(x2+y2)=4\left( x^{2}+y^{2} \right)= 9 and y2=4xy^{2}= 4x intersect at the point QQ. Let an ellipse, centered at the origin OO, has lengths of semi-minor and semi-major axes equal to OQOQ and 6 , respectively. If e and ll respectively denote the eccentricity and the length of the latus rectum of this ellipse, then le2\frac{l}{e^{2}} is equal to

  4. Question 4

    28 Jul 2022

    175
    Two tangent lines l1l_{1} and l2l_{2} are drawn from the point (2,0)(2,0) to the parabola 2y2=−x2y^{2}= - x. If the lines 111_{1} and l2l_{2} are also tangent to the circle (x−5)2+y2=r(x - 5)^{2}+y^{2}= r, then 17r17r is equal to

  5. Question 5

    30 Jan 2023

    164
    Let AA be a point on the xx-axis. Common tangents are drawn from AA to the curves x2+y2=8x^{2}+y^{2}= 8 and y2=16xy^{2}= 16x. If one of these tangents touches the two curves at QQ and RR, then (QR)2(QR)^{2} is equal to

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  6. Question 6

    10 Apr 2023

    84
    Let a common tangent to the curves y2=4xy^{2}= 4x and (x−4)2+y2=16(x - 4)^{2}+y^{2}= 16 touch the curves at the points PP and QQ. Then (PQ)2(PQ)^{2} is equal to