Question

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

• A. 0
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is D

 To solve the problem, let's first understand the sets \(S_n\) and \(T_n\).

The set \(S_n = \{ z \in \mathbb{C} : | z - 3 + 2i | = \frac{n}{4} \}\) represents a circle in the complex plane centered at \(3 - 2i\) with radius \(\frac{n}{4}\).

The set \(T_n = \{ z \in \mathbb{C} : | z - 2 + 3i | = \frac{1}{n} \}\) represents a circle in the complex plane centered at \(2 - 3i\) with radius \(\frac{1}{n}\).

We need to find the integer values of \(n\) for which the intersection \(S_n \cap T_n = \phi\) is empty. This means the two circles must not intersect.

The distance between the centers of the circles is:

\(\text{Distance} = | (3 - 2i) - (2 - 3i) | = | 1 + i | = \sqrt{1^2 + 1^2} = \sqrt{2}\).

For the circles not to intersect, the sum of the radii must be less than the distance between the centers:

\(\frac{n}{4} + \frac{1}{n} < \sqrt{2}\).

Rewriting the inequality:

\(\frac{n^2 + 4}{4n} < \sqrt{2}\).

Cross-multiplying gives:

\(n^2 + 4 < 4n\sqrt{2}\),

\(n^2 - 4n\sqrt{2} + 4 < 0\).

This expression is a quadratic inequality in terms of \(n\). Solving it involves finding the roots of the equation:

\(n^2 - 4n\sqrt{2} + 4 = 0\).

The discriminant for this quadratic is:

\(\Delta = (4\sqrt{2})^2 - 4 \times 1 \times 4 = 32 - 16 = 16\).

The roots are:

\(n = \frac{4\sqrt{2} \pm \sqrt{16}}{2} = \frac{4\sqrt{2} \pm 4}{2}\).

Hence, the roots are:

\(n = 2(\sqrt{2} + 1)\) and \(n = 2(\sqrt{2} - 1)\).

Approximating these, we have:

\(\sqrt{2} \approx 1.414\), so,

\(n \approx 2(2.414) = 4.828\) and \(n \approx 2(0.414) = 0.828\).

Since \(n\) has to be a natural number, possibilities are \(1, 2, 3, 4\).

Therefore, the number of elements in the set \(\{ n \in \mathbb{N} : S_n \cap T_n = \phi \}\) is: 4.