Animated Solution for Mathematics - Complex Numbers: For $n\in \mathbb{N}$, let $S_n=\{z\in \mathbb{C}:|z-(3+2i)|=\frac{n}{4}\}$ and $T_n=\{z\in \mathbb{C}:|z-(2+3i)|=\frac{1}{n}\}$. Then the number of elements in the set $\{n\in \mathbb{N}:S_n\cap T_n=\varnothing \}$ is:

Welcome, fellow traveler of the complex plane! Today, we are going to explore a beautiful problem that bridges the gap between algebra and geometry.

We are looking at two sets, $S_n$ and $T_n$, defined by the modulus of complex numbers. At first glance, these might look like abstract equations, but they are actually the heartbeat of geometry: circles.

Let $S_n=\{z\in \mathbb{C}:|z-(3+2i)|=\frac{n}{4}\}$. This is the definition of a circle in the complex plane with center $C_1(3,2)$ and radius $r_1=\frac{n}{4}$.

Now, consider $T_n=\{z\in \mathbb{C}:|z-(2+3i)|=\frac{1}{n}\}$. This is a second circle centered at $C_2(2,3)$ with radius $r_2=\frac{1}{n}$.

We have a dynamic system where one circle expands and the other contracts as $n$ increases. Our goal is to find the values of $n$ for which these circles never touch.

To understand their interaction, we calculate the distance $d$ between their centers. Using the distance formula between $(3,2)$ and $(2,3)$:

This distance is our anchor; it remains constant regardless of the value of $n$.

For two circles to have an empty intersection, they must be separated. This occurs in two distinct geometric configurations:

1. The circles are completely outside each other: $d>r_1+r_2$. 2. One circle is entirely contained within the other: $d<|r_1-r_2|$.

For Case 1 ($d>r_1+r_2$), we have:

Multiplying by $4n$ (given $n>0$), we obtain $4\sqrt{2}n>n^2+4$, or:

The roots of $n^2-4\sqrt{2}n+4=0$ are $2\sqrt{2}\pm 2$. Approximating these, we find $n\in (0.828,4.828)$. Since $n$ is a natural number, $n\in \{1,2,3,4\}$.

For Case 2 ($d<|r_1-r_2|$), we have:

For $n\ge 2$, this simplifies to $\sqrt{2}<\frac{n}{4}-\frac{1}{n}$, which rearranges to $n^2-4\sqrt{2}n-4>0$. The positive root is $2\sqrt{2}+2\sqrt{3}\approx 6.292$.

Thus, $n>6.292$, giving us the set $n\in \{7,8,9,\dots \}$.

Combining these results, the valid values for $n$ are $\{1,2,3,4,7,8,9,\dots \}$.

Because the second set extends to infinity, the total number of elements is infinite. You have successfully navigated the complex plane and mastered the geometry of circles!