If z is a complex number, then `| 3z -1| = 3 | z-2|` represents

Options

A. Y-axis
B. A circle
C. X-axis
D. A line parallel to y-axis

Answer

To solve the equation \( |3z - 1| = 3|z - 2| \), where \( z \) is a complex number, we can follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Substitute \( z \) into the equation
Substituting \( z \) into the equation gives:
\[ |3(x + iy) - 1| = 3|x + iy - 2| \]
This simplifies to:
\[ |3x + 3iy - 1| = 3|x - 2 + iy| \]

### Step 3: 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 4: Set up the equation
Now we have:
\[ \sqrt{(3x - 1)^2 + (3y)^2} = 3\sqrt{(x - 2)^2 + y^2} \]

### Step 5: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ (3x - 1)^2 + (3y)^2 = 9((x - 2)^2 + y^2) \]

### Step 6: Expand both sides
Expanding both sides:
\[ (3x - 1)^2 + 9y^2 = 9((x - 2)^2 + y^2) \]
This becomes:
\[ 9x^2 - 6x + 1 + 9y^2 = 9(x^2 - 4x + 4 + y^2) \]

### Step 7: Simplify the equation
Expanding the right-hand side:
\[ 9x^2 - 6x + 1 + 9y^2 = 9x^2 - 36x + 36 + 9y^2 \]
Now, cancel \( 9x^2 \) and \( 9y^2 \) from both sides:
\[ -6x + 1 = -36x + 36 \]

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

### Step 9: Interpret the result
Since \( x \) is constant, the equation \( |3z - 1| = 3|z - 2| \) represents a vertical line in the complex plane at \( x = \frac{7}{6} \).

### Conclusion
The final answer is that the equation represents a line parallel to the y-axis.

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