For n ∈ N let \(S_n\)={z∈C:|z−3+2i|=\(\frac{n}{4}\)} and \(T_n\)={z ∈ C:|z−2+3i|=\(\frac{1}{n}\)}.Then the number of elements in the set
- 0
- 2
- Infinite
- 4
The Correct Option is C
Solution and Explanation
\(S_n\)={z∈C:|z−3+2i|=\(\frac{n}{4}\)}
represents a circle with centre C1(3, –2) and radius
r1=\(\frac{n}{4}\)
Similarly,
Tn represents circle with centre C2(2, –3) and radius
r2=\(\frac{1}{n}\)
As Sn ∩ Tn = φ
C1C2>r1 + r2 OR C1C2< |r1 – r2|
\(\sqrt2\)>\(\frac{n}{4}+\frac{1}{n}\)
OR
\(\sqrt2\)<|\(\frac{n}{4}-\frac{1}{n}\)|
n = 1, 2, 3, 4 n may take infinite values.
The Correct Option is (C): Infinite
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Concepts Used:
Sets
In mathematics, a set is a well-defined collection of objects. Sets are named and demonstrated using capital letter. In the set theory, the elements that a set comprises can be any sort of thing: people, numbers, letters of the alphabet, shapes, variables, etc.
Read More: Set Theory
Elements of a Set:
The items existing in a set are commonly known to be either elements or members of a set. The elements of a set are bounded in curly brackets separated by commas.
Read Also: Set Operation
Cardinal Number of a Set:
The cardinal number, cardinality, or order of a set indicates the total number of elements in the set.
Read More: Types of Sets