For a complex number z, if `z^(2)+barz-z=4i` and z does not lie in the first quadrant, then `("where "i^(2)=1)`

Options

A. `|z|=sqrt2`
B. `|z|=2sqrt2`
C. `arg(z)=(-pi)/(4)`
D. `arg(z)=(3pi)/(4)`

Answer

To solve the equation \( z^2 + \bar{z} - z = 4i \) where \( z \) is a complex number and does not lie in the first quadrant, we will follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part of \( z \). The conjugate \( \bar{z} \) is given by:
\[ \bar{z} = x - iy \]

### Step 2: Substitute \( z \) and \( \bar{z} \) into the equation
Substituting \( z \) and \( \bar{z} \) into the equation \( z^2 + \bar{z} - z = 4i \):
\[ (x + iy)^2 + (x - iy) - (x + iy) = 4i \]

### Step 3: Expand \( z^2 \)
Now, we expand \( (x + iy)^2 \):
\[ (x + iy)^2 = x^2 + 2xyi - y^2 = (x^2 - y^2) + 2xyi \]
Thus, substituting this back into the equation gives:
\[ (x^2 - y^2 + 2xyi) + (x - iy) - (x + iy) = 4i \]

### Step 4: Combine like terms
Combining the terms, we have:
\[ (x^2 - y^2 + x - x) + (2xy - y - y)i = 4i \]
This simplifies to:
\[ (x^2 - y^2) + (2xy - 2y)i = 4i \]

### Step 5: Equate real and imaginary parts
From the equation, we can equate the real and imaginary parts:
1. Real part: \( x^2 - y^2 = 0 \)
2. Imaginary part: \( 2xy - 2y = 4 \)

### Step 6: Solve the equations
From the first equation \( x^2 - y^2 = 0 \), we can write:
\[ x^2 = y^2 \implies x = \pm y \]

Substituting \( x = y \) into the second equation:
\[ 2y^2 - 2y = 4 \implies 2y^2 - 2y - 4 = 0 \implies y^2 - y - 2 = 0 \]
Factoring gives:
\[ (y - 2)(y + 1) = 0 \implies y = 2 \text{ or } y = -1 \]

If \( y = 2 \), then \( x = 2 \) or \( x = -2 \). If \( y = -1 \), then \( x = -1 \) or \( x = 1 \).

### Step 7: Determine valid solutions
Since \( z \) does not lie in the first quadrant, we discard \( (2, 2) \) and \( (1, -1) \) as they lie in the first quadrant. Thus, the valid solutions are:
1. \( z = -1 - i \)
2. \( z = -2 + 2i \)

### Step 8: Calculate the modulus of \( z \)
Now, we calculate the modulus \( |z| \) for \( z = -1 - i \):
\[ |z| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 9: Conclusion
The modulus of \( z \) is \( \sqrt{2} \).