We are given the equation involving a complex number $z$:

$\bar{z} + i |z| = 12 + 9i$

where $\bar{z}$ is the conjugate of $z$ and $|z|$ is the modulus of $z$. We need to find the imaginary part of $z$.

Step 1: Express $z$

Let $z = x + yi$, where $x$ is the real part and $y$ is the imaginary part of $z$.

• The conjugate of $z$ is $\bar{z} = x - yi$.
• The modulus of $z$ is $|z| = \sqrt{x^2 + y^2}$.

Step 2: Substitute into the equation

Substituting $\bar{z} = x - yi$ and $|z| = \sqrt{x^2 + y^2}$ into the given equation:

$(x - yi) + i \sqrt{x^2 + y^2} = 12 + 9i$

Step 3: Separate into real and imaginary parts

Now, we can separate the real and imaginary parts on both sides of the equation.

• Real part: $x$
• Imaginary part: $-y + \sqrt{x^2 + y^2}$

So the equation becomes:

$x = 12 \quad \text{(Real part)}$ $-y + \sqrt{x^2 + y^2} = 9 \quad \text{(Imaginary part)}$

Step 4: Solve for $y$

Substitute $x = 12$ into the imaginary part equation:

$-y + \sqrt{12^2 + y^2} = 9$ $-y + \sqrt{144 + y^2} = 9$ Move $y$ to the other side:

$\sqrt{144 + y^2} = y + 9$ Square both sides to eliminate the square root:

$144 + y^2 = (y + 9)^2$ Expand the right side:

$144 + y^2 = y^2 + 18y + 81$ Simplify:

$144 = 18y + 81$ $63 = 18y$ $y = \frac{63}{18} = 3.5$

Step 5: Conclusion

The imaginary part of $z$ is $\boxed{3.5}$.

Let me know if you would like more details or have any questions.

Here are five related questions for further practice:

1. How do you find the modulus of a complex number?
2. What happens when you square a complex number?
3. Can a complex number have a zero real part? What would that mean geometrically?
4. How do you determine the real part of a complex number in a similar equation?
5. How would the equation change if $|z|$ were given directly?

Tip: Always double-check complex equations by separately equating real and imaginary parts, which helps avoid mistakes.