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Slope of a common tangent to an ellipse and a concentric circle

The same question, asked 3 times: 2021 Paper 26*, 26 Jun 2022*, 10 Apr 2023. Every wording below, oldest first.

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  1. Question 1

    2021 Paper 26

    85
    Let LL be a common tangent line to the curves 4x2+9y2=364x^{2}+ 9y^{2}= 36 and (2x)2+(2y)2=31(2x)^{2}+ (2y)^{2}= 31. Then the square of the slope of the line LL is

  2. Question 2

    26 Jun 2022

    161
    If mm is the slope of a common tangent to the curves x216+y29=1\frac{x^{2}}{16}+\frac{y^{2}}{9}= 1 and x2+y2=12x^{2}+y^{2}= 12, then 12m212m^{2} is equal to :

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  3. Question 3

    10 Apr 2023

    160
    Let a circle of radius 4 be concentric to the ellipse 15x2+19y2=28515x^{2}+ 19y^{2}= 285. Then the common tangents are inclined to the minor axis of the ellipse at the angle.

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