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 find 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 endpoints of the diameter of the circle.

### Step 1: Identify the points \( z_1 \) and \( z_2 \)

Given:
- \( z_1 = 3i \) which can be expressed as \( (0, 3) \) in the Cartesian plane.
- \( z_2 = 4 + 3i \) which can be expressed as \( (4, 3) \) in the Cartesian plane.

### Step 2: Calculate the center of the circle

The center \( C \) of the circle can be found using the midpoint formula:
\[ C = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of \( z_1 \) and \( z_2 \):
\[ C = \left( \frac{0 + 4}{2}, \frac{3 + 3}{2} \right) = \left( 2, 3 \right) \]

### Step 3: Calculate the radius of the circle

The radius \( r \) of the circle is half the distance between \( z_1 \) and \( z_2 \). We first calculate the distance \( d \):
\[ d = |z_2 - z_1| = |(4 + 3i) - 3i| = |4 + 3i - 3i| = |4| = 4 \]
Thus, the radius \( r \) is:
\[ r = \frac{d}{2} = \frac{4}{2} = 2 \]

### Step 4: Write the equation of the circle

The standard form of the equation of a circle with center \( (h, k) \) and radius \( r \) is:
\[ |z - C|^2 = r^2 \]
Substituting the center \( C(2, 3) \) and radius \( r = 2 \):
\[ |z - (2 + 3i)|^2 = 2^2 = 4 \]

### Step 5: Relate to the given equation

Now, we relate this back to the original equation:
\[ |z - z_1|^2 + |z - z_2|^2 = k \]
Since we have established that the circle has a radius of \( 2 \), we can derive that:
\[ k = |z_2 - z_1|^2 \]
Calculating \( |z_2 - z_1| \):
\[ |z_2 - z_1| = |(4 + 3i) - 3i| = |4| = 4 \]
Thus, we find:
\[ k = |z_2 - z_1|^2 = 4^2 = 16 \]

### Final Answer

The value of \( k \) is \( \boxed{16} \).