The locus of the points representing the complex numbers z for which `|z|-2=|z-i|-|z+5i|=0`, is

Options

A. a circle with center at the origin
B. a straight line passing through the origin
C. the single point `(0,-2)`
D. none of these

Answer

To solve the problem, we need to find the locus of the points representing the complex numbers \( z \) for which the following conditions hold:

1. \( |z| - 2 = 0 \)
2. \( |z - i| - |z + 5i| = 0 \)

We will analyze these conditions step by step.

### Step 1: Analyze the first condition

The first condition \( |z| - 2 = 0 \) can be rewritten as:

\[ |z| = 2 \]

This represents a circle in the complex plane with a radius of 2 centered at the origin (0, 0).

### Step 2: Analyze the second condition

The second condition \( |z - i| - |z + 5i| = 0 \) can be rewritten as:

\[ |z - i| = |z + 5i| \]

This means that the distance from the point \( z \) to the point \( i \) (which is (0, 1) in the complex plane) is equal to the distance from \( z \) to the point \( -5i \) (which is (0, -5) in the complex plane).

### Step 3: Interpret the second condition geometrically

The equation \( |z - i| = |z + 5i| \) describes the perpendicular bisector of the line segment joining the points \( (0, 1) \) and \( (0, -5) \).

To find the midpoint of these two points:

\[ \text{Midpoint} = \left( 0, \frac{1 + (-5)}{2} \right) = \left( 0, -2 \right) \]

The perpendicular bisector will be a horizontal line (since both points have the same x-coordinate) at the y-coordinate of the midpoint, which is \( y = -2 \).

### Step 4: Combine the conditions

Now we have two conditions:

1. \( |z| = 2 \) (circle of radius 2 centered at the origin)
2. \( y = -2 \) (horizontal line)

### Step 5: Find the intersection of the circle and the line

The circle \( |z| = 2 \) can be expressed in terms of \( x \) and \( y \):

\[ x^2 + y^2 = 4 \]

Substituting \( y = -2 \) into the circle equation:

\[ x^2 + (-2)^2 = 4 \]

This simplifies to:

\[ x^2 + 4 = 4 \]

\[ x^2 = 0 \implies x = 0 \]

### Conclusion

Thus, the only point that satisfies both conditions is:

\[ (0, -2) \]

This means the locus of the points representing the complex numbers \( z \) is the single point \( 0 - 2i \).

### Final Answer

The locus of the points representing the complex numbers \( z \) is the single point \( (0, -2) \).

---