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Two circles touching each other: point of contact and centres

This type of question, asked 4 times: 2021 Paper 21, 25 Jul 2022, 8 Apr 2024, 7 Apr 2025. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 21

    87
    Two circles each of radius 5 units touch each other at the point (1,2)(1,2). If the equation of their common tangent is 4x+3y=104x + 3y = 10, and C1(α,β)C_{1}(\alpha,\beta) and C2(γ,δ)C_{2}(\gamma,\delta), C1≠C2C_{1}\neq C_{2} are their centres, then ∣(α+β)(γ+δ)∣|(\alpha + \beta)(\gamma + \delta)| is equal to

  2. Question 2

    25 Jul 2022

    178
    If the circles x2+y2+6x+8y+16=0x^{2}+y^{2}+ 6x + 8y + 16 = 0 and x2+y2+2(3−3)x+2(4−6)yx^{2}+y^{2}+ 2(3 -\sqrt{3})x + 2(4 -\sqrt{6})y =k+63+86,k>0= k + 6\sqrt{3}+ 8\sqrt{6},k > 0, touch internally at the point P(α,β)P(\alpha,\beta), then (α+3)2+(β+6)2(\alpha +\sqrt{3})^{2}+ (\beta +\sqrt{6})^{2} is equal to

  3. Question 3

    8 Apr 2024

    63
    Let the circles C1:(x−α)2+(y−β)2=r12C_{1}:(x - \alpha)^{2}+ (y - \beta)^{2}=r_{1}^{2} and C2:(x−8)2+(y−152)2=r22C_{2}:(x - 8)^{2}+\left( y -\frac{15}{2} \right)^{2}=r_{2}^{2} touch each other externally at the point (6,6)(6,6). If the point (6,6)(6,6) divides the line segment joining the centres of the circles C1C_{1} and C2C_{2} internally in the ratio 2:12:1, then (α+β)+4(r12+r22)(\alpha + \beta) + 4\left( r_{1}^{2}+r_{2}^{2} \right) equals

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  4. Question 4

    7 Apr 2025

    59
    Let C1C_{1} be the circle in the third quadrant of radius 3 , that touches both coordinate axes. Let C2C_{2} be the circle with centre (1,3)(1,3) that touches C1C_{1} externally at the point (α,β)(\alpha,\beta). If (β−α)2=mn,gcd(m,n)=1(\beta-\alpha)^{2}=\frac{m}{n},gcd(m,n) = 1, then m+nm + n is equal to :

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