The centre of circle represented by `|z + 1| = 2 |z - 1|` in the complex plane is

Options

A. 0
B. `5/3`
C. `1/3`
D. None of these

Answer

To find the center of the circle represented by the equation \( |z + 1| = 2 |z - 1| \) in the complex plane, we will follow these steps:

### Step 1: Rewrite the equation in terms of \( z \)
We start with the given equation:
\[ |z + 1| = 2 |z - 1| \]
Assume \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then we can rewrite the equation as:
\[ |x + iy + 1| = 2 |x + iy - 1| \]

### Step 2: Express the moduli
The left-hand side can be expressed as:
\[ |x + 1 + iy| = \sqrt{(x + 1)^2 + y^2} \]
The right-hand side can be expressed as:
\[ 2 |x - 1 + iy| = 2 \sqrt{(x - 1)^2 + y^2} \]
Thus, we have:
\[ \sqrt{(x + 1)^2 + y^2} = 2 \sqrt{(x - 1)^2 + y^2} \]

### Step 3: Square both sides
Squaring both sides gives:
\[ (x + 1)^2 + y^2 = 4((x - 1)^2 + y^2) \]

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

### Step 5: Rearrange the equation
Rearranging gives:
\[ x^2 + 2x + 1 + y^2 - 4x^2 + 8x - 4 - 4y^2 = 0 \]
Combining like terms results in:
\[ -3x^2 + 10x - 3 - 3y^2 = 0 \]

### Step 6: Multiply through by -1
To simplify, multiply the entire equation by -1:
\[ 3x^2 + 3y^2 - 10x + 3 = 0 \]

### Step 7: Divide by 3
Dividing the entire equation by 3 gives:
\[ x^2 + y^2 - \frac{10}{3}x + 1 = 0 \]

### Step 8: Rewrite in standard form
We can rewrite this in the standard form of a circle:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \( g = -\frac{10}{6} = -\frac{5}{3} \) and \( f = 0 \).

### Step 9: Identify the center
The center of the circle is given by the coordinates \( (-g, -f) \):
\[ \text{Center} = \left(-\left(-\frac{5}{3}\right), -0\right) = \left(\frac{5}{3}, 0\right) \]

### Final Answer
Thus, the center of the circle is:
\[ \boxed{\left(-\frac{5}{3}, 0\right)} \]