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

Options

A. A. 6
B. B. 12
C. C. 18
D. D. 36

Answer

To solve the problem, we need to find the modulus of the complex number \( z \) that satisfies the given conditions. Let's break down the solution step by step.

### Step 1: Analyze the first condition
The first condition given is:
\[ \left| \frac{z - 4}{z - 8} \right| = 1 \]
This implies that the distance from \( z \) to 4 is equal to the distance from \( z \) to 8. Therefore, \( z \) lies on the perpendicular bisector of the line segment joining the points 4 and 8 on the real axis.

### Step 2: Find the equation of the perpendicular bisector
The midpoint of the segment joining 4 and 8 is:
\[ \text{Midpoint} = \frac{4 + 8}{2} = 6 \]
The perpendicular bisector is a vertical line at \( x = 6 \). Therefore, we can express \( z \) as:
\[ z = 6 + iy \quad \text{(where \( y \) is any real number)} \]

### Step 3: Analyze the second condition
The second condition given is:
\[ \left| \frac{z}{z - 2} \right| = \frac{3}{2} \]
Substituting \( z = 6 + iy \):
\[ \left| \frac{6 + iy}{(6 + iy) - 2} \right| = \frac{3}{2} \]
This simplifies to:
\[ \left| \frac{6 + iy}{4 + iy} \right| = \frac{3}{2} \]

### Step 4: Separate the modulus
Using the property of modulus, we can write:
\[ \frac{|6 + iy|}{|4 + iy|} = \frac{3}{2} \]
Calculating the moduli:
\[ |6 + iy| = \sqrt{6^2 + y^2} = \sqrt{36 + y^2} \]
\[ |4 + iy| = \sqrt{4^2 + y^2} = \sqrt{16 + y^2} \]
Thus, we have:
\[ \frac{\sqrt{36 + y^2}}{\sqrt{16 + y^2}} = \frac{3}{2} \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives:
\[ 2\sqrt{36 + y^2} = 3\sqrt{16 + y^2} \]
Squaring both sides:
\[ 4(36 + y^2) = 9(16 + y^2) \]
Expanding both sides:
\[ 144 + 4y^2 = 144 + 9y^2 \]
Rearranging gives:
\[ 4y^2 - 9y^2 = 0 \implies -5y^2 = 0 \implies y^2 = 0 \]
Thus, \( y = 0 \).

### Step 6: Find \( z \)
Substituting \( y = 0 \) back into the expression for \( z \):
\[ z = 6 + 0i = 6 \]

### Step 7: Calculate \( |z| \)
Finally, we find the modulus of \( z \):
\[ |z| = |6| = 6 \]

### Conclusion
The value of \( |z| \) is:
\[ \boxed{6} \]