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Condition for two circles to touch: find the parameter

This type of question, asked 3 times: 10 May 2024 S1, 12 May 2024 S2, 17 Apr 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    10 May 2024 S1

    146
    Circles x2+y2=9x^2+y^2=9 and x2+y2+2αx+2y+1=0x^2+y^2+2\alpha x+2y+1=0 touch internally. Value of α3\alpha^3 is

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  2. Question 2

    12 May 2024 S2

    148
    The circles x2+y2+2ax+c=0x^2+y^2+2ax+c=0 and x2+y2+2by+c=0x^2+y^2+2by+c=0 touch each other externally, if

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  3. Question 3

    17 Apr 2026 S2

    56
    If the circles x2+y2=16x^2 + y^2 = 16 and x2+y2+2ax+4y+4=0x^2 + y^2 + 2ax + 4y + 4 = 0 touch each other internally then a=a =

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