if z is a complex number such that z+|z|=8+12i, then the value of `|z^(2)|` is equal to

Options

A. 228
B. 144
C. 121
D. 169

Answer

To solve the problem, we start with the equation given:

\[ z + |z| = 8 + 12i \]

1. **Express \( z \) in terms of its real and imaginary parts:**
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. The modulus of \( z \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]

2. **Substituting into the equation:**
Substitute \( z \) and \( |z| \) into the equation:
\[ (x + iy) + \sqrt{x^2 + y^2} = 8 + 12i \]

3. **Separate real and imaginary parts:**
This gives us two equations by comparing the real and imaginary parts:
- Real part: \( x + \sqrt{x^2 + y^2} = 8 \)
- Imaginary part: \( y = 12 \)

4. **Substituting \( y \) into the real part equation:**
Substitute \( y = 12 \) into the real part equation:
\[ x + \sqrt{x^2 + 12^2} = 8 \]
\[ x + \sqrt{x^2 + 144} = 8 \]

5. **Isolate the square root:**
Rearranging gives:
\[ \sqrt{x^2 + 144} = 8 - x \]

6. **Square both sides:**
Squaring both sides results in:
\[ x^2 + 144 = (8 - x)^2 \]
\[ x^2 + 144 = 64 - 16x + x^2 \]

7. **Simplify the equation:**
Cancel \( x^2 \) from both sides:
\[ 144 = 64 - 16x \]
Rearranging gives:
\[ 16x = 64 - 144 \]
\[ 16x = -80 \]
\[ x = -5 \]

8. **Determine \( z \):**
Now we have \( x = -5 \) and \( y = 12 \), so:
\[ z = -5 + 12i \]

9. **Calculate \( |z| \):**
The modulus of \( z \) is:
\[ |z| = \sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

10. **Calculate \( z^2 \):**
Now we calculate \( z^2 \):
\[ z^2 = (-5 + 12i)^2 = (-5)^2 + 2(-5)(12i) + (12i)^2 \]
\[ = 25 - 120i - 144 \]
\[ = -119 - 120i \]

11. **Calculate \( |z^2| \):**
Finally, we find the modulus of \( z^2 \):
\[ |z^2| = \sqrt{(-119)^2 + (-120)^2} \]
\[ = \sqrt{14161 + 14400} = \sqrt{28561} = 169 \]

Thus, the value of \( |z^2| \) is:

\[ \boxed{169} \]