If z = x + iy , then | 3z - 1| = 3 | z - 2| represents

Options

A. x- axis
B. y - axis
C. circle
D. `x = 7/6`

Answer

To solve the equation \( |3z - 1| = 3 |z - 2| \) where \( z = x + iy \), we will follow these steps:

### Step 1: Substitute \( z \) into the equation
Given \( z = x + iy \), we can substitute this into the equation:
\[ |3(x + iy) - 1| = 3 |(x + iy) - 2| \]
This simplifies to:
\[ |3x + 3iy - 1| = 3 |x - 2 + iy| \]

### Step 2: Simplify both sides
The left-hand side becomes:
\[ |3x - 1 + 3iy| = \sqrt{(3x - 1)^2 + (3y)^2} \]
The right-hand side becomes:
\[ 3 |(x - 2) + iy| = 3 \sqrt{(x - 2)^2 + y^2} \]

### Step 3: Set up the equation
Now we have:
\[ \sqrt{(3x - 1)^2 + (3y)^2} = 3 \sqrt{(x - 2)^2 + y^2} \]

### Step 4: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ (3x - 1)^2 + (3y)^2 = 9((x - 2)^2 + y^2) \]

### Step 5: Expand both sides
Expanding the left-hand side:
\[ (3x - 1)^2 = 9x^2 - 6x + 1 \]
\[ (3y)^2 = 9y^2 \]
So, the left-hand side becomes:
\[ 9x^2 - 6x + 1 + 9y^2 \]

Expanding the right-hand side:
\[ 9((x - 2)^2 + y^2) = 9(x^2 - 4x + 4 + y^2) = 9x^2 - 36x + 36 + 9y^2 \]

### Step 6: Set the expanded forms equal to each other
Now we have:
\[ 9x^2 - 6x + 1 + 9y^2 = 9x^2 - 36x + 36 + 9y^2 \]

### Step 7: Cancel out common terms
Cancelling \( 9x^2 \) and \( 9y^2 \) from both sides:
\[ -6x + 1 = -36x + 36 \]

### Step 8: Solve for \( x \)
Rearranging gives:
\[ -6x + 36x = 36 - 1 \]
\[ 30x = 35 \]
\[ x = \frac{35}{30} = \frac{7}{6} \]

### Step 9: Conclusion
The value of \( x \) is \( \frac{7}{6} \). The equation \( |3z - 1| = 3 |z - 2| \) represents a specific relationship in the complex plane, which can be interpreted geometrically.