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Question

There are two circles whose equations are x2+y2=4 and x2+y224x10y+a2=0,aϵZ. If the two circle have exactly two common tangents then the number of possible values of a is

A
2
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B
8
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C
13
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D
none of these
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Solution

The correct option is C 13
Equation of circles are x2+y2=4 ...(1)
and, x2+y224x10y+a2=0 ...(2)
Center of (1) is C1(0,0) and radius r1=2
Center of C2(12,5) and radius r2=169a2
d= distance between centers =C1C2=144+25=13
If the two circles have exactly two common tangents, then
169a2>0 and r1+r2>d
(a13)(a+13)<0 and 2+169a2>13
13<a<13 and 169a2>121
13<a<13 and a248<0
13<a<13 and 48<a<48
48<a<48
Since a is an integer
a6,5,4,...,4,5,6
Therefore the number of possible values of a is 13

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