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Question

If two circles of radii 5 units touches each other at (1,2) and the equation of the common tangent is 4x+3y=10, then the equation of the circle is/are

A
x2+y210x10y+25=0
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B
x2+y2+6x+2y15=0
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C
x2+y210x10y25=0
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D
x2+y2+6x2y+15=0
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Solution

The correct option is B x2+y2+6x+2y15=0
Let the centre of the circle be (h,k)
Given tangent is 4x+3y=10
It passes through (1,2)
Line passing through (1,2) and centre is perpendicular to the given tangent, so
k2h1=34(1)

Now, the radius of circle
(h1)2+(k2)2=5(h1)2+(k2)2=25
Using equation (1), we get
(h1)2+916(h1)2=25(h1)2=16h1=±4h=5,3k=5,1

Hence, the circles are
x2+y210x10y+25=0 and
x2+y2+6x+2y15=0

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