Key Concepts and Formulas

• Complex Number Representation: A complex number $z$ can be represented as $z = x + iy$, where $x = \text{Re}(z)$ is the real part and $y = \text{Im}(z)$ is the imaginary part. The modulus is $|z| = \sqrt{x^2 + y^2}$.
• Argument of a Complex Number: The argument of a complex number $z = x + iy$ is the angle $\theta$ such that $\tan(\theta) = \frac{y}{x}$. $\arg(z)$ is the angle the vector from the origin to $z$ makes with the positive real axis.
• Locus of Points: An equation involving a complex variable $z$ can define a curve or region in the complex plane.
• Parabola Equation: The general form of a horizontal parabola is $(y-k)^2 = 4p(x-h)$, where $(h,k)$ is the vertex.

Step-by-Step Solution

Step 1: Express the Locus Condition in terms of x and y

We are given $|z - 3| = \text{Re}(z)$. Let $z = x + iy$. We want to express this equation in terms of $x$ and $y$ to find the locus.

$|x + iy - 3| = x$ $|(x - 3) + iy| = x$ $\sqrt{(x - 3)^2 + y^2} = x$

Squaring both sides:

$(x - 3)^2 + y^2 = x^2$ $x^2 - 6x + 9 + y^2 = x^2$ $y^2 = 6x - 9$ $y^2 = 6\left(x - \frac{3}{2}\right)$

This is a parabola opening to the right with vertex at $\left(\frac{3}{2}, 0\right)$.

Step 2: Express the Argument Condition in terms of x and y

We are given $\arg(z_1 - z_2) = \frac{\pi}{4}$. Let $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$. We want to find a relationship between $x_1, y_1, x_2,$ and $y_2$.

$z_1 - z_2 = (x_1 - x_2) + i(y_1 - y_2)$ $\arg(z_1 - z_2) = \arg((x_1 - x_2) + i(y_1 - y_2)) = \frac{\pi}{4}$

Since $\arg((x_1 - x_2) + i(y_1 - y_2)) = \frac{\pi}{4}$, we have:

$\tan\left(\frac{\pi}{4}\right) = \frac{y_1 - y_2}{x_1 - x_2}$ $1 = \frac{y_1 - y_2}{x_1 - x_2}$ $x_1 - x_2 = y_1 - y_2$

Step 3: Apply the Locus Equation to z1 and z2

Since $z_1$ and $z_2$ satisfy the equation $y^2 = 6x - 9$, we have:

$y_1^2 = 6x_1 - 9 \quad \text{... (1)}$ $y_2^2 = 6x_2 - 9 \quad \text{... (2)}$

Step 4: Combine the Equations to Solve for y1 + y2

We want to find $y_1 + y_2$. Subtract equation (2) from equation (1):

$y_1^2 - y_2^2 = 6x_1 - 6x_2$ $(y_1 - y_2)(y_1 + y_2) = 6(x_1 - x_2)$

From Step 2, we know that $x_1 - x_2 = y_1 - y_2$. Substitute this into the equation:

$(y_1 - y_2)(y_1 + y_2) = 6(y_1 - y_2)$

Now, we consider the case where $y_1 - y_2 = 0$. If $y_1 = y_2$, then $x_1 = x_2$, so $z_1 = z_2$. If $z_1 = z_2$, then $z_1 - z_2 = 0$, and $\arg(0)$ is undefined, which contradicts the given condition $\arg(z_1 - z_2) = \frac{\pi}{4}$. Therefore, $y_1 - y_2 \neq 0$.

Since $y_1 - y_2 \neq 0$, we can divide both sides by $y_1 - y_2$:

$y_1 + y_2 = 6$

The imaginary part of $z_1 + z_2$ is $y_1 + y_2$.

Therefore, the imaginary part of $z_1 + z_2$ is $6$.

Common Mistakes & Tips

• Dividing by zero: Always check if you are dividing by zero when a variable is in the denominator.
• Geometric intuition: Visualizing complex numbers as points in the plane can help understand the argument condition.
• Algebraic manipulation: Be careful when expanding and factoring equations.

Summary

By expressing the locus condition as an equation of a parabola and using the argument condition to relate the real and imaginary parts of $z_1$ and $z_2$, we were able to set up a system of equations. Solving these equations allowed us to find the imaginary part of $z_1 + z_2$. The imaginary part of $z_1 + z_2$ is $6$.

Final Answer

The final answer is \boxed{6}.