If the equation `|z-z_1|^2 + |z-z_2|^2 =k`, represents the equation of a circle, where `z_1 = 3i, z_2 = 4+3i` are the extremities of a diameter, then the value of k is

Options

A. `1//4`
B. 4
C. 1
D. None of these

Answer

To solve the problem, we need to determine the value of \( k \) in the equation \( |z - z_1|^2 + |z - z_2|^2 = k \), where \( z_1 = 3i \) and \( z_2 = 4 + 3i \). The points \( z_1 \) and \( z_2 \) are the extremities of a diameter of the circle.

### Step-by-Step Solution:

1. **Identify the Points**:
- We have \( z_1 = 3i \) and \( z_2 = 4 + 3i \).

2. **Calculate the Distance Between \( z_1 \) and \( z_2 \)**:
- We find \( |z_1 - z_2| \):
\[ z_1 - z_2 = 3i - (4 + 3i) = 3i - 4 - 3i = -4 \]
- The modulus is:
\[ |z_1 - z_2| = |-4| = 4 \]

3. **Determine the Radius of the Circle**:
- The distance \( |z_1 - z_2| \) represents the diameter of the circle. Therefore, the radius \( r \) is:
\[ r = \frac{|z_1 - z_2|}{2} = \frac{4}{2} = 2 \]

4. **Calculate \( k \)**:
- The equation \( |z - z_1|^2 + |z - z_2|^2 = k \) represents a circle, and for a circle, \( k \) is given by:
\[ k = 2r^2 + |z_1|^2 + |z_2|^2 \]
- First, calculate \( |z_1|^2 \) and \( |z_2|^2 \):
\[ |z_1|^2 = |3i|^2 = 3^2 = 9 \]
\[ |z_2|^2 = |4 + 3i|^2 = 4^2 + 3^2 = 16 + 9 = 25 \]
- Now substitute the values into the equation for \( k \):
\[ k = 2(2^2) + 9 + 25 = 2(4) + 9 + 25 = 8 + 9 + 25 = 42 \]

5. **Final Result**:
- Thus, the value of \( k \) is:
\[ \boxed{42} \]