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 :
Solution
$S_{n}=\left\{z \in C:|z-3+2 i|=\frac{n}{4}\right\}$ represents a circle with centre $C_{1}(3,-2)$ and radius $r_{1}=\frac{n}{4}$
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Similarly $T_{n}$ represents circle with centre $C_{2}(2,-3)$ and radius $r_{2}=\frac{1}{n}$
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$\text { As } S_{n} \cap T_{n}=\phi$<br/><br/>
$$
\begin{array}{llr}
C_{1} C_{2}>r_{1}+r_{2}& \text { OR } & C_{1} C_{2}<\left|r_{1}-r_{2}\right| \\\\
\sqrt{2}>\frac{n}{4}+\frac{1}{n} & \text { OR } & \sqrt{2}<\left|\frac{n}{4}-\frac{1}{n}\right| \\\\
n=1,2,3,4&& n \text { may take infinite values }
\end{array}
$$
About this question
Subject: Mathematics · Chapter: Complex Numbers and Quadratic Equations · Topic: Complex Numbers and Argand Plane
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