If z satisfies the equation `((z-2)/(z+2))((barz-2)/(barz+2))=1`, then minimum value of `|z|` is equal to :

Options

A. 0
B. 2
C. 4
D. 6

Explanation

To solve the equation

\[ \frac{(z-2)}{(z+2)} \cdot \frac{(\bar{z}-2)}{(\bar{z}+2)} = 1, \]

we will follow these steps:

### Step 1: Substitute \( z \) with \( x + iy \)
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part of the complex number.

### Step 2: Write the conjugate of \( z \)
The conjugate of \( z \) is given by \( \bar{z} = x - iy \).

### Step 3: Substitute \( z \) and \( \bar{z} \) into the equation
Substituting \( z \) and \( \bar{z} \) into the equation gives:

\[ \frac{(x + iy - 2)}{(x + iy + 2)} \cdot \frac{(x - iy - 2)}{(x - iy + 2)} = 1. \]

### Step 4: Simplify the fractions
This can be rewritten as:

\[ \frac{(x - 2 + iy)}{(x + 2 + iy)} \cdot \frac{(x - 2 - iy)}{(x + 2 - iy)} = 1. \]

### Step 5: Cross-multiply
Cross-multiplying gives:

\[ (x - 2 + iy)(x - 2 - iy) = (x + 2 + iy)(x + 2 - iy). \]

### Step 6: Expand both sides
The left-hand side expands to:

\[ (x - 2)^2 + y^2, \]

and the right-hand side expands to:

\[ (x + 2)^2 + y^2. \]

### Step 7: Set the equations equal
Setting the two expansions equal gives:

\[ (x - 2)^2 + y^2 = (x + 2)^2 + y^2. \]

### Step 8: Cancel \( y^2 \)
Since \( y^2 \) appears on both sides, we can cancel it:

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

### Step 9: Expand and simplify
Expanding both sides results in:

\[ x^2 - 4x + 4 = x^2 + 4x + 4. \]

### Step 10: Cancel \( x^2 \) and simplify further
Cancelling \( x^2 \) from both sides gives:

\[ -4x + 4 = 4x + 4. \]

### Step 11: Rearrange the equation
Rearranging leads to:

\[ -4x - 4x = 4 - 4, \]

which simplifies to:

\[ -8x = 0. \]

### Step 12: Solve for \( x \)
Thus, we find:

\[ x = 0. \]

### Step 13: Find the minimum value of \( |z| \)
Since \( x = 0 \), \( z \) can be expressed as \( z = iy \). The modulus of \( z \) is given by:

\[ |z| = |iy| = |y|. \]

### Step 14: Determine the minimum value
The minimum value of \( |y| \) is 0, which occurs when \( y = 0 \). Therefore, the minimum value of \( |z| \) is:

\[ \boxed{0}. \]