Consider the circle C0: x^2+y^2-2x-4y-4=0. Match the number of common tangents of C0 with each circle in List-I to a value in List-II. List-I (P)…

$C_0$ has centre $(1,2)$ and radius $3$. Compare the distance $d$ of centres with $r_1\pm r_2$.
(P) Centre $(4,6)$, radius $2$: $d=5=3+2$, externally tangent, so $3$ common tangents: (4).
(Q) Centre $(1,4)$, radius $1$: $d=2=3-1$, internally tangent, so $1$ common tangent: (2).
(R) Centre $(2,3)$, radius $1$: $d=\sqrt2<3-1=2$, one circle lies inside the other, so $0$ tangents: (1).
(S) Centre $(9,2)$, radius $2$: $d=8>3+2$, circles are apart, so $4$ common tangents: (5).
The correct code is P→4, Q→2, R→1, S→5. The code with S→3 mistakes disjoint circles for intersecting ones, and the code with P→5 treats external tangency as separation.