If the real part of `(bar(z) + 2)/( bar(z)-1)` is 4, then the locus of the point representing z in the complex plane is

Options

A. A circle
B. A parabola
C. A hyperbola
D. An ellipse

Answer

To solve the problem, we need to find the locus of the point representing \( z \) in the complex plane given that the real part of \( \frac{\bar{z} + 2}{\bar{z} - 1} \) is 4.

Let’s denote \( z = x + iy \), where \( x \) and \( y \) are the real and imaginary parts of \( z \), respectively. The conjugate of \( z \) is \( \bar{z} = x - iy \).

### Step 1: Substitute \( \bar{z} \) into the expression

We start with the expression:

\[ \frac{\bar{z} + 2}{\bar{z} - 1} = \frac{(x - iy) + 2}{(x - iy) - 1} = \frac{x + 2 - iy}{x - 1 - iy} \]

### Step 2: Separate the real and imaginary parts

To separate the real and imaginary parts, we multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{(x + 2 - iy)(x - 1 + iy)}{(x - 1 - iy)(x - 1 + iy)} \]

Calculating the denominator:

\[ (x - 1)^2 + y^2 \]

Calculating the numerator:

\[ (x + 2)(x - 1) + (x + 2)(iy) - (iy)(x - 1) - y^2 \]

This simplifies to:

\[ (x^2 + x - 2 + y^2) + i(y(x + 2 - (x - 1))) = (x^2 + x - 2 + y^2) + i(3y) \]

Thus, we have:

\[ \frac{(x^2 + x - 2 + y^2) + i(3y)}{(x - 1)^2 + y^2} \]

### Step 3: Find the real part

The real part of the expression is:

\[ \text{Re}\left(\frac{(x^2 + x - 2 + y^2) + i(3y)}{(x - 1)^2 + y^2}\right) = \frac{x^2 + x - 2 + y^2}{(x - 1)^2 + y^2} \]

### Step 4: Set the real part equal to 4

We set the real part equal to 4:

\[ \frac{x^2 + x - 2 + y^2}{(x - 1)^2 + y^2} = 4 \]

Cross-multiplying gives:

\[ x^2 + x - 2 + y^2 = 4((x - 1)^2 + y^2) \]

### Step 5: Expand and simplify

Expanding the right side:

\[ x^2 + x - 2 + y^2 = 4(x^2 - 2x + 1 + y^2) \]

This simplifies to:

\[ x^2 + x - 2 + y^2 = 4x^2 - 8x + 4 + 4y^2 \]

Rearranging gives:

\[ 0 = 3x^2 + 3y^2 - 9x + 6 \]

### Step 6: Divide by 3

Dividing the entire equation by 3:

\[ 0 = x^2 + y^2 - 3x + 2 \]

### Step 7: Complete the square

Completing the square for \( x \):

\[ 0 = (x - \frac{3}{2})^2 + y^2 - \frac{9}{4} + 2 \]

This leads to:

\[ (x - \frac{3}{2})^2 + y^2 = \frac{1}{4} \]

### Conclusion

The equation represents a circle with center \( \left(\frac{3}{2}, 0\right) \) and radius \( \frac{1}{2} \).

Thus, the locus of the point representing \( z \) in the complex plane is a **circle**.