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

Options

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

Explanation

To solve the problem step by step, we will analyze the given equations involving the complex number \( z \).

### Step 1: Understand the given conditions
We have two conditions:
1. \( \left| \frac{z-4}{z-8} \right| = 1 \)
2. \( \left| \frac{z}{z-2} \right| = \frac{3}{2} \)

### Step 2: Analyze the first condition
The first condition \( \left| \frac{z-4}{z-8} \right| = 1 \) implies that the magnitudes of the numerator and denominator are equal:
\[ |z - 4| = |z - 8| \]

### Step 3: Set up the equation
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then:
\[ |z - 4| = |(x - 4) + iy| = \sqrt{(x - 4)^2 + y^2} \]
\[ |z - 8| = |(x - 8) + iy| = \sqrt{(x - 8)^2 + y^2} \]
Setting these equal gives:
\[ \sqrt{(x - 4)^2 + y^2} = \sqrt{(x - 8)^2 + y^2} \]

### Step 4: Square both sides
Squaring both sides eliminates the square roots:
\[ (x - 4)^2 + y^2 = (x - 8)^2 + y^2 \]
The \( y^2 \) terms cancel out, simplifying to:
\[ (x - 4)^2 = (x - 8)^2 \]

### Step 5: Expand and simplify
Expanding both sides:
\[ x^2 - 8x + 16 = x^2 - 16x + 64 \]
Cancel \( x^2 \) from both sides:
\[ -8x + 16 = -16x + 64 \]
Rearranging gives:
\[ 8x = 48 \quad \Rightarrow \quad x = 6 \]

### Step 6: Analyze the second condition
Now we move to the second condition \( \left| \frac{z}{z-2} \right| = \frac{3}{2} \):
This implies:
\[ |z| = \frac{3}{2} |z - 2| \]

### Step 7: Substitute \( z = x + iy \)
Substituting \( z = 6 + iy \):
\[ |6 + iy| = \sqrt{6^2 + y^2} = \sqrt{36 + y^2} \]
\[ |z - 2| = |(6 - 2) + iy| = |4 + iy| = \sqrt{4^2 + y^2} = \sqrt{16 + y^2} \]
Thus, we have:
\[ \sqrt{36 + y^2} = \frac{3}{2} \sqrt{16 + y^2} \]

### Step 8: Square both sides again
Squaring both sides gives:
\[ 36 + y^2 = \frac{9}{4}(16 + y^2) \]
Multiplying through by 4 to eliminate the fraction:
\[ 4(36 + y^2) = 9(16 + y^2) \]
This simplifies to:
\[ 144 + 4y^2 = 144 + 9y^2 \]

### Step 9: Rearranging the equation
Rearranging gives:
\[ 4y^2 - 9y^2 = 0 \quad \Rightarrow \quad -5y^2 = 0 \]
Thus, \( y^2 = 0 \) which implies \( y = 0 \).

### Step 10: Find \( |z| \)
Now we have \( z = 6 + 0i \). The modulus is:
\[ |z| = \sqrt{6^2 + 0^2} = \sqrt{36} = 6 \]

### Final Answer
Thus, \( |z| = 6 \).

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