For [n N], let [S_n=z C:|z-3+2 i|=n4] and [T_n=z C:|z-2+3 i|=1n].

Question

For $$\mathrm{n} \in \mathbf{N}$$, let $$\mathrm{S}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-3+2 i|=\frac{\mathrm{n}}{4}\right\}$$ and $$\mathrm{T}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-2+3 i|=\frac{1}{\mathrm{n}}\right\}$$. Then the number of elements in the set $$\left\{n \in \mathbf{N}: S_{n} \cap T_{n}=\phi\right\}$$ is :

Accepted Answer

$S_{n}=\left\{z \in C:|z-3+2 i|=\frac{n}{4}\right\}$ represents a circle with centre $C_{1}(3,-2)$ and radius $r_{1}=\frac{n}{4}$

Similarly $T_{n}$ represents circle with centre $C_{2}(2,-3)$ and radius $r_{2}=\frac{1}{n}$

$\text { As } S_{n} \cap T_{n}=\phi$

$$
\[\begin{gathered} C_{1} C_{2}>r_{1}+r_{2}\quad  \text { OR } \quad  C_{1} C_{2}\frac{n}{4}+\frac{1}{n} \quad  \text { OR } \quad  \sqrt{2}<\left|\frac{n}{4}-\frac{1}{n}\right| \ n=1,2,3,4\quad \quad  n \text { may take infinite values } \end{gathered}$$
\]

Solutions · 1