The region in the Argand diagram defined by `|z-2i|+|z+2i| lt 5` is the ellipse with major axis along

Options

A. the real axis
B. the imaginary axis
C. `y=x`
D. `y=-x`

Explanation

To solve the problem, we need to analyze the given inequality in the context of complex numbers and the Argand diagram. The inequality provided is:

\[ |z - 2i| + |z + 2i| < 5 \]

### Step 1: Understanding the Expression
The expression \( |z - 2i| + |z + 2i| \) represents the sum of distances from the point \( z \) in the complex plane to the points \( 2i \) and \( -2i \).

### Step 2: Identifying the Foci
The points \( 2i \) and \( -2i \) can be represented in Cartesian coordinates as:
- \( F_1 = (0, 2) \)
- \( F_2 = (0, -2) \)

These points are the foci of the ellipse.

### Step 3: Recognizing the Ellipse Condition
The inequality \( |z - 2i| + |z + 2i| < 5 \) indicates that we are looking for the interior of an ellipse. The general definition of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant. In this case, the constant is \( 5 \).

### Step 4: Determining the Major Axis
Since the foci \( (0, 2) \) and \( (0, -2) \) are aligned along the imaginary axis (the y-axis), the major axis of the ellipse will also be along the imaginary axis.

### Step 5: Conclusion
Thus, the region defined by the inequality \( |z - 2i| + |z + 2i| < 5 \) is an ellipse with its major axis along the imaginary axis.

### Final Answer
The major axis of the ellipse is along the imaginary axis.

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