Complex number z satisfy the equation `|z-(4/z)|=2`
The difference between the least and the greatest moduli of complex number is (a) 2 (b) 4 (c) 1 (d) 3

Options

A. 2
B. 4
C. 1
D. 3

Explanation

To solve the problem, we need to analyze the equation given by the complex number \( z \) that satisfies the condition \( |z - \frac{4}{z}| = 2 \).

### Step 1: Rewrite the equation
We start with the equation:
\[ |z - \frac{4}{z}| = 2 \]
This implies that the expression \( z - \frac{4}{z} \) can be thought of as a vector in the complex plane, and its magnitude is 2.

### Step 2: Use the triangle inequality
Using the triangle inequality, we can express:
\[ |z| - \left| \frac{4}{z} \right| \leq |z - \frac{4}{z}| \leq |z| + \left| \frac{4}{z} \right| \]
Let \( r = |z| \). Then \( \left| \frac{4}{z} \right| = \frac{4}{|z|} = \frac{4}{r} \).

### Step 3: Set up inequalities
Substituting into the inequalities gives us:
\[ r - \frac{4}{r} \leq 2 \quad \text{(1)} \]
\[ r + \frac{4}{r} \geq 2 \quad \text{(2)} \]

### Step 4: Solve inequality (1)
From inequality (1):
\[ r - \frac{4}{r} \leq 2 \]
Multiplying through by \( r \) (assuming \( r > 0 \)):
\[ r^2 - 2r - 4 \leq 0 \]
Using the quadratic formula to find the roots:
\[ r = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 + 16}}{2} = \frac{2 \pm \sqrt{20}}{2} = 1 \pm \sqrt{5} \]
Thus, the roots are \( 1 + \sqrt{5} \) and \( 1 - \sqrt{5} \). Since \( r \) must be positive, we consider:
\[ 1 - \sqrt{5} < 0 \quad \text{(not valid)} \]
Thus, we focus on:
\[ r \leq 1 + \sqrt{5} \]

### Step 5: Solve inequality (2)
From inequality (2):
\[ r + \frac{4}{r} \geq 2 \]
Multiplying through by \( r \):
\[ r^2 - 2r + 4 \geq 0 \]
This is a quadratic that is always positive (discriminant \( < 0 \)), so it does not provide any additional restrictions on \( r \).

### Step 6: Combine results
From the inequalities, we have:
\[ r \geq 1 - \sqrt{5} \quad \text{(not valid since it's negative)} \]
And:
\[ r \leq 1 + \sqrt{5} \]

### Step 7: Determine the range of \( r \)
Thus, the valid range for \( r \) (the modulus of \( z \)) is:
\[ r \in [1 - \sqrt{5}, 1 + \sqrt{5}] \]
However, since \( 1 - \sqrt{5} < 0 \), we only consider:
\[ r \in [0, 1 + \sqrt{5}] \]

### Step 8: Calculate the difference between the least and greatest moduli
The least modulus is \( 0 \) and the greatest modulus is \( 1 + \sqrt{5} \). Therefore, the difference is:
\[ (1 + \sqrt{5}) - 0 = 1 + \sqrt{5} \]

### Step 9: Final calculation
The difference between the least and greatest moduli:
\[ 1 + \sqrt{5} - (1 - \sqrt{5}) = 2 \]

### Conclusion
Thus, the difference between the least and greatest moduli of the complex number \( z \) is:
\[ \boxed{2} \]