The number of points in the complex plane that satisfy the conditions `| z- 2| = 2 sqrt(2) , z ( 1- i) + bar(z) (1 + i) = 4 ` is

Options

A. 0
B. 1
C. 2
D. more than 2

Answer

To solve the problem, we need to find the number of points in the complex plane that satisfy the given conditions:

1. \( |z - 2| = 2\sqrt{2} \)
2. \( z(1 - i) + \overline{z}(1 + i) = 4 \)

Let's denote the complex number \( z \) as \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 1: Analyze the first condition
The first condition \( |z - 2| = 2\sqrt{2} \) can be interpreted as the equation of a circle in the complex plane centered at \( (2, 0) \) with a radius of \( 2\sqrt{2} \).

### Step 2: Analyze the second condition
The second condition involves \( z(1 - i) + \overline{z}(1 + i) = 4 \). We can rewrite this using \( z = x + iy \) and \( \overline{z} = x - iy \):

\[ (x + iy)(1 - i) + (x - iy)(1 + i) = 4 \]

### Step 3: Expand the second condition
Expanding the left-hand side:

\[ (x + iy)(1 - i) = x(1) + x(-i) + iy(1) + iy(-i) = x - ix + iy + y = (x + y) + i(y - x) \]

\[ (x - iy)(1 + i) = x(1) + x(i) - iy(1) - iy(i) = x + ix - iy - y = (x - y) + i(x - y) \]

Now combine both parts:

\[ ((x + y) + i(y - x)) + ((x - y) + i(x + y)) = 4 \]

### Step 4: Combine real and imaginary parts
Combining the real parts and the imaginary parts gives:

Real part: \( (x + y) + (x - y) = 2x \)

Imaginary part: \( (y - x) + (x + y) = 2y \)

Thus, we have:

\[ 2x + 2y = 4 \]

Dividing through by 2:

\[ x + y = 2 \quad \text{(Equation 1)} \]

### Step 5: Substitute \( y \) in terms of \( x \)
From Equation 1, we can express \( y \) in terms of \( x \):

\[ y = 2 - x \]

### Step 6: Substitute into the first condition
Now substitute \( y = 2 - x \) into the first condition \( |z - 2| = 2\sqrt{2} \):

\[ |x + i(2 - x) - 2| = 2\sqrt{2} \]

This simplifies to:

\[ |x - 2 + i(2 - x)| = 2\sqrt{2} \]

Taking the modulus:

\[ \sqrt{(x - 2)^2 + (2 - x)^2} = 2\sqrt{2} \]

### Step 7: Simplify the modulus equation
Squaring both sides:

\[ (x - 2)^2 + (2 - x)^2 = 8 \]

Expanding both terms:

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

Adding these gives:

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

This simplifies to:

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

### Step 8: Solve the quadratic equation
Subtracting 8 from both sides:

\[ 2x^2 - 8x = 0 \]

Factoring out \( 2x \):

\[ 2x(x - 4) = 0 \]

This gives us:

\[ x = 0 \quad \text{or} \quad x = 4 \]

### Step 9: Find corresponding \( y \) values
For \( x = 0 \):

\[ y = 2 - 0 = 2 \quad \Rightarrow \quad (0, 2) \]

For \( x = 4 \):

\[ y = 2 - 4 = -2 \quad \Rightarrow \quad (4, -2) \]

### Conclusion
Thus, the points that satisfy both conditions are \( (0, 2) \) and \( (4, -2) \). Therefore, the number of points in the complex plane that satisfy the conditions is:

**Answer: 2 points**