Interpret the sets geometrically

$S_n=\{z\in \mathbb C:|z-(3-2i)|=\tfrac n4\}$ is a circle with center $C_1=3-2i\quad \Rightarrow \quad (3,-2)$ and radius $r_1=\frac n4.$

Similarly, $T_n=\{z\in \mathbb C:|z-(2-3i)|=\tfrac1n\}$ is a circle with center $C_2=2-3i\quad \Rightarrow \quad (2,-3)$ and radius $r_2=\frac1n.$

Find distance between centers

$d=| (3-2i)-(2-3i)|=|1+i|=\sqrt{2}.$

Condition for two circles to have empty intersection

Two circles do not intersect if either

• they are externally disjoint: $d>r_1+r_2,$
• or one lies completely inside the other without touching: $d<|r_1-r_2|.$

So we check both cases.

Check the possibility $d<|r_1-r_2|$

Here $|r_1-r_2|=\left|\frac n4-\frac1n\right|.$

For small and moderate natural $n$, this is much smaller than $\sqrt2$ except possibly larger $n$. Let us see directly:

• For $n=1$: $|r_1-r_2|=\left|\frac14-1\right|=\frac34<\sqrt2.$
• For $n\ge 2$, $\frac n4-\frac1n$ increases, but if this exceeds $\sqrt2$, then certainly also $r_1+r_2>\sqrt2$ and the circles will overlap or contain one another depending on exact values. We only need values where intersection is empty.

The more relevant disjoint condition here is external disjointness, since radii are positive and center distance is fixed.

Use external disjointness condition

We need $\sqrt2>\frac n4+\frac1n.$

Multiply by $4n>0$: $4\sqrt2\,n>n^2+4.$

Rearranging, $n^2-4\sqrt2\,n+4<0.$

Solve the quadratic equation $n^2-4\sqrt2\,n+4=0.$

Its roots are

Check these values

• $n=1$: $r_1+r_2=\frac14+1=1.25<\sqrt2,$ so disjoint.
• $n=2$: $r_1+r_2=\frac12+\frac12=1<\sqrt2,$ so disjoint.
• $n=3$: $r_1+r_2=\frac34+\frac13=\frac{13}{12}\approx1.083<\sqrt2,$ so disjoint.
• $n=4$: $r_1+r_2=1+\frac14=1.25<\sqrt2,$ so disjoint.

For $n\ge 5$, $\frac n4+\frac1n\ge \frac54+\frac15=1.45>\sqrt2,$ so they are not externally disjoint.

Also, checking internal disjointness for $n\ge5$: $\left|\frac n4-\frac1n\right|<\sqrt2$ fails to make them disjoint in the required range of natural numbers here, so no extra values arise.

Count the values

The set is $\{1,2,3,4\},$ so the number of elements is $4.$

Option matching

Correct option is: $\boxed{\text{D}}$