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Complex Numbers Part 2

Section: Mathematics

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Question : 98 of 106

Marks: +1, -0

For

n \in N

S_n=\{z\in C: |z-3+2i|=\frac{n}{4}\}

T_n=\{z\in C: |z-2+3i|=\frac{1}{n}\}

Then the number of elements in the set

\{n\in N: S_n\cap T_n=\phi\}

[25-Jul-2022-Shift-1]

0
2
3
4

Solution: 👈: Video Solution

S_n=\{z\in C: |z-3+2i|=\frac{n}{4}\}

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}

As

S_n\cap T_n=\phi

C_1C_2>r_1+r_2\;\;

OR

\;\;C_1C_2<|r_1-r_2|

\sqrt{2}>\frac{n}{4}+\frac{1}{n}\;\;

OR

\;\;\sqrt{2} < \left|\frac{n}{4}-\frac{1}{n}\right|

n=1,2,3,4\;\;n

may take infinite values

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