If `a gt 0` and the equation `|z-a^(2)|+|z-2a|=3`, represents an ellipse, then 'a' belongs to the interval

Options

A. (1,3)
B. `(sqrt(2),sqrt(3))`
C. (0,3)
D. `(1,sqrt(3))`

Answer

To solve the problem, we need to analyze the equation given and determine the conditions under which it represents an ellipse. The equation is:

\[ |z - a^2| + |z - 2a| = 3 \]

where \( z \) is a complex number and \( a > 0 \).

### Step-by-Step Solution:

1. **Substituting \( z \)**:
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then, we can express the equation in terms of \( x \) and \( y \):
\[ |z - a^2| = |(x - a^2) + iy| = \sqrt{(x - a^2)^2 + y^2} \]
\[ |z - 2a| = |(x - 2a) + iy| = \sqrt{(x - 2a)^2 + y^2} \]
Thus, the equation becomes:
\[ \sqrt{(x - a^2)^2 + y^2} + \sqrt{(x - 2a)^2 + y^2} = 3 \]

2. **Understanding the Geometric Interpretation**:
The equation represents the sum of distances from the points \( (a^2, 0) \) and \( (2a, 0) \) to the point \( (x, y) \). This is the definition of an ellipse, where the sum of distances from two foci is constant.

3. **Finding the Foci**:
The foci of the ellipse are at \( (a^2, 0) \) and \( (2a, 0) \). The distance between the foci is:
\[ d = |2a - a^2| = |2a - a^2| \]

4. **Condition for an Ellipse**:
For the figure to be an ellipse, the sum of the distances (which is 3) must be greater than the distance between the foci:
\[ 3 > |2a - a^2| \]

5. **Analyzing the Expression**:
We need to solve the inequality:
\[ 3 > |2a - a^2| \]
This can be split into two cases:
- Case 1: \( 2a - a^2 < 3 \)
- Case 2: \( 2a - a^2 > -3 \)

6. **Solving Case 1**:
\[ 2a - a^2 < 3 \implies -a^2 + 2a - 3 < 0 \implies a^2 - 2a + 3 > 0 \]
The roots of the equation \( a^2 - 2a + 3 = 0 \) are complex, indicating that this quadratic is always positive.

7. **Solving Case 2**:
\[ 2a - a^2 > -3 \implies -a^2 + 2a + 3 > 0 \implies a^2 - 2a - 3 < 0 \]
Factoring gives:
\[ (a - 3)(a + 1) < 0 \]
The solution to this inequality is:
\[ -1 < a < 3 \]

8. **Considering \( a > 0 \)**:
Since \( a > 0 \), we restrict our solution to:
\[ 0 < a < 3 \]

### Final Answer:
Thus, the value of \( a \) belongs to the interval:
\[ (0, 3) \]