Let z be a complex number satisfying
`|(z-4)/(z-8)|=1and|(z)/(z-2)|=3/2`
What is `|(z-6)/(z+6)|` equal to ?

Options

A. 3
B. 2
C. 1
D. 0

Explanation

To solve the problem, we need to find the value of \( \left| \frac{z-6}{z+6} \right| \) given the conditions:

1. \( \left| \frac{z-4}{z-8} \right| = 1 \)
2. \( \left| \frac{z}{z-2} \right| = \frac{3}{2} \)

Let's go through the solution step by step.

### Step 1: Analyze the first condition
From the first condition, \( \left| \frac{z-4}{z-8} \right| = 1 \), we can deduce that the magnitudes of the numerator and denominator are equal:

\[ |z - 4| = |z - 8| \]

This means that the complex number \( z \) lies on the perpendicular bisector of the points \( 4 \) and \( 8 \) on the real line. The midpoint of \( 4 \) and \( 8 \) is \( 6 \), so we can express this condition as:

\[ |z - 6| = |z - 6| \quad \text{(which is trivially true)} \]

### Step 2: Analyze the second condition
From the second condition, \( \left| \frac{z}{z-2} \right| = \frac{3}{2} \), we can separate the magnitudes:

\[ |z| = \frac{3}{2} |z - 2| \]

### Step 3: Substitute \( z = x + iy \)
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

The first condition gives us:

\[ |z - 4| = |z - 8| \implies \sqrt{(x - 4)^2 + y^2} = \sqrt{(x - 8)^2 + y^2} \]

Squaring both sides:

\[ (x - 4)^2 + y^2 = (x - 8)^2 + y^2 \]

Cancelling \( y^2 \) from both sides:

\[ (x - 4)^2 = (x - 8)^2 \]

Expanding both sides:

\[ x^2 - 8x + 16 = x^2 - 16x + 64 \]

Simplifying:

\[ 8x - 48 = 0 \implies x = 6 \]

### Step 4: Substitute \( x = 6 \) into the second condition
Now substitute \( x = 6 \) into the second condition:

\[ |z| = \frac{3}{2} |z - 2| \]

Calculating \( |z| \):

\[ |z| = |6 + iy| = \sqrt{6^2 + y^2} = \sqrt{36 + y^2} \]

Calculating \( |z - 2| \):

\[ |z - 2| = |(6 - 2) + iy| = |4 + iy| = \sqrt{4^2 + y^2} = \sqrt{16 + y^2} \]

Now substituting into the second condition:

\[ \sqrt{36 + y^2} = \frac{3}{2} \sqrt{16 + y^2} \]

### Step 5: Square both sides
Squaring both sides:

\[ 36 + y^2 = \frac{9}{4}(16 + y^2) \]

Multiplying through by \( 4 \):

\[ 144 + 4y^2 = 9(16 + y^2) \]

Expanding:

\[ 144 + 4y^2 = 144 + 9y^2 \]

Rearranging gives:

\[ 0 = 5y^2 \implies y^2 = 0 \implies y = 0 \]

### Step 6: Conclusion for \( z \)
Thus, we have:

\[ z = 6 + 0i = 6 \]

### Step 7: Calculate \( \left| \frac{z-6}{z+6} \right| \)
Now we need to find:

\[ \left| \frac{z - 6}{z + 6} \right| = \left| \frac{6 - 6}{6 + 6} \right| = \left| \frac{0}{12} \right| = 0 \]

### Final Answer
Thus, the value of \( \left| \frac{z-6}{z+6} \right| \) is:

\[ \boxed{0} \]