Complex numbers z satisfy the equaiton `|z-(4//z)|=2`
Locus of z if `|z-z_(1)| = |z-z_(2)|`, where `z_(1)` and `z_(2)` are complex numbers with the greatest and the least moduli, is

Options

A. line parallel to the real axis
B. line parallel to the imaginary axis
C. line having a positive slope
D. line having a negative slope

Explanation

To solve the problem, we need to analyze the given equation involving complex numbers and determine the locus of the complex number \( z \).

### Step 1: Start with the given equation
The equation given is:
\[ |z - \frac{4}{z}| = 2 \]

### Step 2: Rewrite the equation
We can rewrite the equation in terms of \( z \) and its conjugate \( \bar{z} \). Multiplying both sides by \( |z| \) gives:
\[ |z| \cdot |z - \frac{4}{z}| = 2|z| \]
This leads us to:
\[ |z|^2 - 4 = 2|z| \]

### Step 3: Square both sides
Squaring both sides of the original equation, we have:
\[ |z - \frac{4}{z}|^2 = 4 \]
Expanding this gives:
\[ (z - \frac{4}{z})(\bar{z} - \frac{4}{\bar{z}}) = 4 \]

### Step 4: Expand the left-hand side
Expanding the left-hand side:
\[ z\bar{z} - 4\left(z + \bar{z}\right) + \frac{16}{z\bar{z}} = 4 \]
Let \( |z|^2 = r^2 \), then \( z\bar{z} = r^2 \). The equation becomes:
\[ r^2 - 4(z + \bar{z}) + \frac{16}{r^2} = 4 \]

### Step 5: Simplify the equation
Rearranging gives:
\[ r^2 - 4(z + \bar{z}) + \frac{16}{r^2} - 4 = 0 \]
This can be simplified further to find the relationship between \( z \) and its modulus \( r \).

### Step 6: Identify the locus
To find the locus, we need to determine the values of \( z \) such that the distance from \( z \) to the points \( z_1 \) and \( z_2 \) (which have the greatest and least moduli) is equal. This is represented by:
\[ |z - z_1| = |z - z_2| \]
This represents the perpendicular bisector of the line segment joining \( z_1 \) and \( z_2 \).

### Step 7: Determine the values of \( z_1 \) and \( z_2 \)
From our previous steps, we find that:
- The maximum modulus \( r_{\text{max}} = \sqrt{5} + 1 \)
- The minimum modulus \( r_{\text{min}} = \sqrt{5} - 1 \)

### Step 8: Conclusion about the locus
The locus of the points \( z \) satisfying the condition \( |z - z_1| = |z - z_2| \) is the perpendicular bisector of the segment joining \( z_1 \) and \( z_2 \). Since \( z_1 \) and \( z_2 \) lie on the real axis, the locus is a vertical line in the complex plane.

### Final Answer
The locus of \( z \) is a line parallel to the imaginary axis.

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