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=r2−r1, 1 common tangents
(iii) If circles do not touch each other, 4 common tangents.
Given equations of circles are
x2+y2−4x−6y−12=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+81−26)=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.