A.
x2+y2+2x+8y−23=0 C1(−1,−4),r1=2√10 x2+y2−4x−10y+19=0C2(2,5),r2=√10
C1C2=√32+92=√90=3√10=r1+r2=3√10
∴3 tangents
B.
x2+y2=1,C1(0,0),r1=1 x2+y2−2x−6y+6=0,C2(1,3),r2=2
C1C2=√1+9=√10>r1+r2>3
∴4 tangents
C. x2+y2−8x+2y=0,C1(4,−2),r1=√17
x2+y2−2x−16y+25=0,C2(1,8),r2=2√10
C1C2=√(3)2+(10)2=√9+100=√109<√17+2√10
2 tangent
D.
x2+y2=4,C1(0,0),r1=2 x2+y2−2x=0,C2(1,0),r2=1 C1C2=1,|r1−r2|=1⇒C1C2=|r1−r2|
∴ One common tangents