Equation of circle `Re(z^2)+2(img(z))^2+2Re(z)=0` where `z=x+iy` .A line passes through the vetex of parabola `x^2-6x+y+13=0` and center of circle, then the y intercept of the line is _____?

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Convert the equation of the circle
The given equation of the circle is:
\[ \text{Re}(z^2) + 2(\text{Im}(z))^2 + 2\text{Re}(z) = 0 \]
where \( z = x + iy \).

First, we need to express \( z^2 \):
\[ z^2 = (x + iy)^2 = x^2 - y^2 + 2xyi \]
Thus,
\[ \text{Re}(z^2) = x^2 - y^2 \quad \text{and} \quad \text{Im}(z^2) = 2xy \]

Now substituting these into the equation:
\[ x^2 - y^2 + 2y^2 + 2x = 0 \]
This simplifies to:
\[ x^2 + y^2 + 2x = 0 \]

### Step 2: Rewrite the equation in standard form
We can complete the square for the \( x \) terms:
\[ x^2 + 2x + y^2 = 0 \implies (x + 1)^2 - 1 + y^2 = 0 \]
This gives:
\[ (x + 1)^2 + y^2 = 1 \]
This is the equation of a circle with center at \( (-1, 0) \) and radius \( 1 \).

### Step 3: Find the vertex of the parabola
The parabola is given by:
\[ x^2 - 6x + y + 13 = 0 \]
Rearranging gives:
\[ y = -x^2 + 6x - 13 \]
To find the vertex, we can complete the square for the \( x \) terms:
\[ y = -\left(x^2 - 6x\right) - 13 \]
Completing the square:
\[ y = -\left((x - 3)^2 - 9\right) - 13 = - (x - 3)^2 + 9 - 13 = - (x - 3)^2 - 4 \]
Thus, the vertex of the parabola is at \( (3, -4) \).

### Step 4: Find the equation of the line
We need to find the slope of the line that passes through the center of the circle \( (-1, 0) \) and the vertex of the parabola \( (3, -4) \).

The slope \( m \) of the line through these two points is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 0}{3 - (-1)} = \frac{-4}{4} = -1 \]

### Step 5: Write the equation of the line
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Using point \( (-1, 0) \):
\[ y - 0 = -1(x + 1) \implies y = -x - 1 \]

### Step 6: Find the y-intercept
To find the y-intercept, set \( x = 0 \):
\[ y = -0 - 1 = -1 \]

### Final Answer
The y-intercept of the line is:
\[ \boxed{-1} \]