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Question

Which of the following point lies on the common tangent at the circle x2+y24x6y36=0 and x2+y210x+2y+22=0.

A
(10,14)
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B
(14,10)
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C
(12,2)
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D
(2,12)
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Solution

The correct option is A (10,14)
C1:(x2)2+(y3)2=72, c1:(2,3)C2:(x5)2+(y+1)2=22, c2:(5,1)

c1c2=(25)2+(3+1)2=5c1c2=r1r2=72=5


Therefore C2 lies inside the C1 and touches only at one point.
Therefore, equation of tangent is,
C1C2=03x4y=29
From the given options
(10,14) lies on the common tangent.

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