If `|z+3i|+|z-i|=8`, then the locus of z, in the Argand plane, is

Options

A. an ellipse of eccentricity `1/2` and major axis along x-axis.
B. an ellipse of eccentricity `1/2` and major axis of along y-axis.
C. an ellipse of eccentricity `1/sqrt(2)` and major axis along y-axis
D. none of these

Explanation

To solve the problem, we need to find the locus of the complex number \( z \) in the Argand plane given the equation \( |z + 3i| + |z - i| = 8 \).

### Step-by-Step Solution:

1. **Identify Points in the Complex Plane**:
- Let \( z = x + yi \), where \( x \) and \( y \) are real numbers.
- The expression \( |z + 3i| \) represents the distance from the point \( (x, y) \) to the point \( (0, -3) \) in the Argand plane.
- The expression \( |z - i| \) represents the distance from the point \( (x, y) \) to the point \( (0, 1) \).

2. **Rewrite the Given Equation**:
- The equation \( |z + 3i| + |z - i| = 8 \) can be interpreted as the sum of distances from the point \( (x, y) \) to the points \( (0, -3) \) and \( (0, 1) \).

3. **Identify the Foci**:
- Let \( S = (0, -3) \) and \( S' = (0, 1) \). These are the foci of the locus.

4. **Interpret the Equation**:
- The equation \( |z + 3i| + |z - i| = 8 \) describes an ellipse with foci at \( S \) and \( S' \). The sum of the distances from any point on the ellipse to the two foci is constant (in this case, 8).

5. **Find the Major Axis**:
- The distance between the foci \( S \) and \( S' \) is \( |1 - (-3)| = 4 \).
- The major axis length \( 2a \) is given as 8, hence \( a = 4 \).

6. **Calculate the Distance Between the Foci**:
- The distance \( 2c \) between the foci is 4, so \( c = 2 \).

7. **Use the Relationship in Ellipses**:
- For an ellipse, the relationship between \( a \), \( b \), and \( c \) is given by \( c^2 = a^2 - b^2 \).
- Here, \( c = 2 \) and \( a = 4 \), thus:
\[ 2^2 = 4^2 - b^2 \implies 4 = 16 - b^2 \implies b^2 = 12 \implies b = 2\sqrt{3} \]

8. **Conclusion**:
- The locus of \( z \) is an ellipse centered at the midpoint of the foci, which is at \( (0, -1) \) with semi-major axis \( a = 4 \) and semi-minor axis \( b = 2\sqrt{3} \).

### Final Answer:
The locus of \( z \) in the Argand plane is an ellipse with foci at \( (0, -3) \) and \( (0, 1) \).

---