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Question

The number of common tangents to the circles x2+y24x6y12=0 and x2+y2+6x+18y+26=0 is

A
1
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B
2
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C
3
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D
4
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Solution

The correct option is C 3
Number of common tangents depend on the position of the circle with respect to each other.
(i) If circles touch externally C1C2=r1+r2, 3 common tangents.
(ii) If circles touch internally C1C2=r2r1, 1 common tangents
(iii) If circles do not touch each other, 4 common tangents.
Given equations of circles are
x2+y24x6y12=0 .....(i)
x2+y2+6x+18y+26=0 ....(ii)
Centre of circle (i) is C1(2,3) and radius, r1
=4+9+12=5
Centre of circle (ii) is C2(3,9) and radius, r2
=9+8126)=8
Now, C1C2=(2+3)2+(3+9)2 C1C2=52+122 C1C2=25+144=13 r1+r2=5+8=13
Also, C1C2=r1+r2
Thus, both circles touch each other externally. Hence, there are three common tangents.

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