If `|Z-2|=2|Z-1|`, then the value of `(Re(Z))/(|Z|^(2))` is (where Z is a complex number and `Re(Z)` represents the real part of Z)

Answer

To solve the equation \( |Z - 2| = 2 |Z - 1| \) where \( Z \) is a complex number, we can express \( Z \) in terms of its real and imaginary parts. Let's denote \( Z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part.

### Step 1: Rewrite the equation in terms of \( x \) and \( y \)
The equation can be rewritten as:
\[ |Z - 2| = |(x + iy) - 2| = |(x - 2) + iy| = \sqrt{(x - 2)^2 + y^2} \]
\[ |Z - 1| = |(x + iy) - 1| = |(x - 1) + iy| = \sqrt{(x - 1)^2 + y^2} \]
Thus, the equation becomes:
\[ \sqrt{(x - 2)^2 + y^2} = 2 \sqrt{(x - 1)^2 + y^2} \]

### Step 2: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ (x - 2)^2 + y^2 = 4((x - 1)^2 + y^2) \]

### Step 3: Expand both sides
Expanding both sides:
\[ (x^2 - 4x + 4 + y^2) = 4(x^2 - 2x + 1 + y^2) \]
This simplifies to:
\[ x^2 - 4x + 4 + y^2 = 4x^2 - 8x + 4 + 4y^2 \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ x^2 - 4x + 4 + y^2 - 4x^2 + 8x - 4 - 4y^2 = 0 \]
Combining like terms results in:
\[ -3x^2 + 4x - 3y^2 = 0 \]
or:
\[ 3x^2 + 3y^2 - 4x = 0 \]

### Step 5: Factor the equation
Factoring out 3 gives:
\[ 3(x^2 + y^2) = 4x \]
Thus:
\[ x^2 + y^2 = \frac{4}{3}x \]

### Step 6: Find the value of \( \frac{\text{Re}(Z)}{|Z|^2} \)
We know that \( |Z|^2 = x^2 + y^2 \). From the equation \( x^2 + y^2 = \frac{4}{3}x \), we can substitute:
\[ \frac{\text{Re}(Z)}{|Z|^2} = \frac{x}{x^2 + y^2} = \frac{x}{\frac{4}{3}x} = \frac{3}{4} \]

### Final Result
Thus, the value of \( \frac{\text{Re}(Z)}{|Z|^2} \) is:
\[ \frac{3}{4} \]