Let A, B, C be three complex numbers as defined below:A = {z : Im z ≥ 1}B = {z : |z - 2 - i| = 3}C = {z : Re((1 - i)z) = 3√2}The number of elements in the set A ∩ B ∩ C isCorrect answer is '1'. Can you explain this answer?

Question

Answer

Given Information:
Three complex numbers are defined as follows:
A = {z : Im z ≥ 1}
B = {z : |z - 2 - i| = 3}
C = {z : Re((1 - i)z) = 3√2}

Explanation:

To find the number of elements in the set A ∩ B ∩ C, we need to find the common complex numbers that satisfy all three conditions.

Step 1: Analyzing Set A
Set A is defined as {z : Im z ≥ 1}, which means the imaginary part of z is greater than or equal to 1. This represents a half-plane above the line y = 1 on the complex plane.

Step 2: Analyzing Set B
Set B is defined as {z : |z - 2 - i| = 3}, which means the distance between z and the point (2, 1) is 3. This represents a circle with center (2, 1) and radius 3 on the complex plane.

Step 3: Analyzing Set C
Set C is defined as {z : Re((1 - i)z) = 3√2}, which means the real part of ((1 - i)z) is equal to 3√2. Simplifying this equation, we get Re(z - iz) = 3√2, which further simplifies to Re(z) - Im(z) = 3√2. This represents a line in the complex plane.

Step 4: Finding the Common Elements
To find the common elements, we need to find the intersection of the half-plane, circle, and line.

The line defined by Set C intersects the circle defined by Set B at a single point. Let's call this point P.

Since the line intersects the circle at only one point, it means that the line does not intersect the circle at any other point.

Now, we need to check if this point P satisfies the condition of Set A. Since the imaginary part of P is equal to 1, it satisfies the condition of Set A.

Therefore, the number of elements in the set A ∩ B ∩ C is 1, which is the single common complex number that satisfies all three conditions.

Final Answer: The number of elements in the set A ∩ B ∩ C is 1.

Answer

A = |z : |mz ≥ 1|

B = {z : |z - 2 - i| = 3 }

C = {z : Re ((1 - i)2 = √2}

Let z = x + iy

Number of element in set A ∩ B ∩ C from the graph = 1

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