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Question

If a circle C passing through (4,0) touches the circle x2+y2+4x−6y−12=0 externally at a point (1,−1), then the radius of the circle C is:

A
57
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B
25
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C
4
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D
5
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Solution

The correct option is C 5
Letthe equation of circle be,
(xh)2+(yk)2=r2
(h,k) is centre and r is radius
and, (yh)2+k2=r2
(lh)2+(l+k)2=r2
Now, the centre of circle C will be lie on the line passing through point of contact and centre of the circle x2+y2+4xby12=0
which has an equation
y+1x1=+3+121
3y3=4x4
4x+3y=1
(h,k) satisfies this equation so,
4h+3k=1
so, 3k=14h
k=14h3
so,
(4h)2+(14h3)2=r2
16+h28h+1+16h28h9=r2
25ah280h9+1459=r2.......(1)
and, (1h)2+(1+14h3)2=r2
1+h22h+(44h3)2=r2
1+h22h+16+16h232h9=r2
25h2950h9+259=r2......(2)
solving (1) & (2) we get,
25h2950h9+259=25h2980h9+1459
30h9=1209
h=4
so, k=1163=5
so, r=[(44)2+(5)2]1/2
r=5

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