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A tangent to one curve also touches a circle: find the constant or the point of contact

This type of question, asked 4 times: 3 May 2023 S2, 19 Apr 2025 S2, 18 Apr 2026 S1, 13 May 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    3 May 2023 S2

    127
    The tangent to the circle x2+y2=5x^{2}+y^{2}=5 at (1,−2)(1,-2) also touches the circle x2+y2−8x+6y+20=0x^{2}+y^{2}-8x+6y+20=0 then the co-ordinates of the corresponding point of contact is

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  2. Question 2

    19 Apr 2025 S2

    147
    If the tangent at (1,7)(1,7) to the curve x2=y−6x^{2}=y- 6 touches the circle x2+y2+16x+12y+C=0x^{2}+y^{2}+ 16x+ 12y+ C = 0, then C=C =

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  3. Question 3

    18 Apr 2026 S1

    57
    The tangent to the circle x2+y2=10x^2 + y^2 = 10 at the point (3,1)(3, 1) touches the circle x2+y2−210 x−20y+k=0x^2 + y^2 - 2\sqrt{10}\,x - 20y + k = 0, then the value of kk is...

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  4. Question 4

    13 May 2026 S2

    63
    If the tangent to the circle x2+y2=5x^2 + y^2 = 5 at point (1,−2)(1, -2) also touches the circle x2+y2−8x+6y+k=0x^2 + y^2 - 8x + 6y + k = 0, then k5=⋯⋯⋯\dfrac{k}{5} = \cdots\cdots\cdots

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