The equation `|z-i|+|z+i|=k, k gt 0` can represent an ellipse, if k=

Options

A. 1
B. 2
C. 4
D. none of these

Explanation

To solve the problem, we need to determine the value of \( k \) for which the equation \( |z - i| + |z + i| = k \) represents an ellipse. Here’s the step-by-step solution:

### Step 1: Understand the Equation
The equation \( |z - i| + |z + i| = k \) represents the sum of distances from the point \( z \) to the points \( i \) and \( -i \) in the complex plane. For the equation to represent an ellipse, the sum of the distances must be greater than the distance between the two fixed points.

### Step 2: Identify the Fixed Points
The fixed points are:
- \( S = i \) which corresponds to the coordinates \( (0, 1) \)
- \( S' = -i \) which corresponds to the coordinates \( (0, -1) \)

### Step 3: Calculate the Distance Between the Fixed Points
To find the distance between points \( S \) and \( S' \):
\[ \text{Distance} = |i - (-i)| = |i + i| = |2i| = 2 \]
Thus, the distance between \( S \) and \( S' \) is \( 2 \).

### Step 4: Establish the Condition for \( k \)
For the equation to represent an ellipse, the value of \( k \) must be greater than the distance between the two points:
\[ k > 2 \]

### Step 5: Identify the Possible Values of \( k \)
Since \( k \) must be greater than \( 2 \), we look for options that satisfy this condition. If the options provided include \( 4 \), then:
\[ k = 4 \quad \text{(which is greater than 2)} \]

### Conclusion
Thus, the value of \( k \) for which the equation \( |z - i| + |z + i| = k \) represents an ellipse is:
\[ \boxed{4} \]