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Step-by-step Solution
Step 1: Rewrite the Circle in Cartesian Form
Given: Re( z^2 ) + 2(Im( z )) ^2 + 2Re( z ) = 0, where z = x + iy .
We know that z^2 = (x + iy)^2 = x^2 - y^2 + 2ixy . Hence:
Re( z^2 ) = x^2 - y^2 , and Im( z ) = y .
Substitute these into the given equation:
x^2 - y^2 + 2y^2 + 2x = 0
⟹ x^2 + y^2 + 2x = 0 .
Step 2: Identify the Center of the Circle
The standard form of a circle is (x + a)^2 + (y + b)^2 = r^2 . We rearrange:
x^2 + 2x + y^2 = 0 ⟹ (x^2 + 2x + 1) + y^2 = 1 .
So, (x + 1)^2 + y^2 = 1 .
This shows the center of the circle is at (-1, 0) .
Step 3: Rewrite the Parabola and Identify its Vertex
The equation of the parabola is:
x^2 - 6x - y + 13 = 0 .
Rearrange in a standard form:
x^2 - 6x = y - 13 .
Complete the square for x :
x^2 - 6x + 9 = y - 13 + 9
(x - 3)^2 = y - 4 .
Therefore, the vertex of the parabola is (3, 4) .
Step 4: Find the Equation of the Line through Both Points
We have two points: the center of the circle (-1, 0) and the vertex of the parabola (3, 4) .
The slope m of the line through these points is:
m = \frac{4 - 0}{3 - (-1)} = \frac{4}{4} = 1 .
Using point-slope form, y - y_1 = m(x - x_1) , and choosing the center (-1, 0) as (x_1, y_1) :
y - 0 = 1 \cdot (x - (-1)) ⟹ y = x + 1 .
Step 5: Determine the y-intercept
In the equation y = x + 1 , the constant term (1) is the y-intercept. Hence, the y-intercept is:
1.
