Key Concepts and Formulas

• Complex Numbers: For $z = x + iy$, $Re(z) = x$, $Im(z) = y$, and $z^2 = (x+iy)^2 = x^2 - y^2 + 2ixy$.
• Circle Equation: Standard form: $(x-h)^2 + (y-k)^2 = r^2$, with center $(h, k)$. General form: $x^2 + y^2 + 2gx + 2fy + c = 0$, center $(-g, -f)$.
• Parabola Equation: Standard form: $(x-h)^2 = 4a(y-k)$, with vertex $(h, k)$.
• Line Equation: Point-slope form: $y - y_1 = m(x - x_1)$, where $m = \frac{y_2 - y_1}{x_2 - x_1}$ is the slope.
• y-intercept: The value of $y$ when $x=0$.

Step-by-Step Solution

Step 1: Finding the Center of the Circle

We are given the equation $Re(z^2) + 2(Im(z))^2 + 2Re(z) = 0$. Our goal is to convert this into Cartesian coordinates to find the center of the circle.

• Substitute $z = x + iy$ into the equation. $z^2 = (x + iy)^2 = x^2 + 2ixy - y^2$ $Re(z^2) = x^2 - y^2$ $Im(z) = y$ $Re(z) = x$

• Substitute these expressions into the given equation: $(x^2 - y^2) + 2(y^2) + 2x = 0$

• Simplify the equation: $x^2 - y^2 + 2y^2 + 2x = 0$ $x^2 + y^2 + 2x = 0$

• Complete the square to get the standard form of the circle equation: $(x^2 + 2x) + y^2 = 0$ $(x^2 + 2x + 1) + y^2 = 1$ $(x + 1)^2 + y^2 = 1$ $(x - (-1))^2 + (y - 0)^2 = 1^2$

• Identify the center of the circle by comparing to the standard form $(x-h)^2 + (y-k)^2 = r^2$. The center of the circle is $C(-1, 0)$.

Step 2: Finding the Vertex of the Parabola

We are given the equation $x^2 - 6x - y + 13 = 0$. Our goal is to convert this into the standard form of a parabola to find the vertex.

• Isolate the $y$ term: $y = x^2 - 6x + 13$

• Complete the square to get the standard form of the parabola equation: $y = (x^2 - 6x) + 13$ $y = (x^2 - 6x + 9) + 13 - 9$ $y = (x - 3)^2 + 4$

• Rewrite in the standard form $(x-h)^2 = 4a(y-k)$: $(x - 3)^2 = y - 4$

• Identify the vertex of the parabola by comparing to the standard form $(x-h)^2 = 4a(y-k)$. The vertex of the parabola is $V(3, 4)$.

Step 3: Finding the Equation of the Line

We need to find the equation of the line passing through the center of the circle $C(-1, 0)$ and the vertex of the parabola $V(3, 4)$.

• Calculate the slope of the line: $m = \frac{4 - 0}{3 - (-1)} = \frac{4}{4} = 1$

• Use the point-slope form of the line equation with the point $C(-1, 0)$ and the slope $m = 1$: $y - 0 = 1(x - (-1))$ $y = x + 1$

Step 4: Finding the y-intercept

We need to find the y-intercept of the line $y = x + 1$.

• Set $x = 0$ and solve for $y$: $y = 0 + 1$ $y = 1$

• The y-intercept is $1$.

Common Mistakes & Tips

• Complex Number Expansion: Be careful when expanding $(x+iy)^2$. The correct expansion is $x^2 + 2ixy - y^2$.
• Completing the Square: Ensure correct arithmetic when completing the square. For example, in the parabola equation, add and subtract 9 correctly.
• Sign Errors: Pay close attention to signs when calculating the slope and using the point-slope form.

Summary

We found the center of the circle by converting the complex equation to Cartesian coordinates and completing the square. We then found the vertex of the parabola by completing the square. Using these two points, we calculated the slope and equation of the line, and finally, we determined the y-intercept by setting x=0. The final y-intercept of the line is 1.

Final Answer The final answer is \boxed{1}.