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To find the equation of the line containing the centers of the two circles, we first need to determine the centers of each circle from their given equations.

### Step 1: Determine the centers of the circles.

The general form of the equation of a circle is:
\[
x^2 + y^2 + Dx + Ey + F = 0
\]
The center \((h, k)\) of the circle can be found using:
\[
h = -\frac{D}{2}, \quad k = -\frac{E}{2}
\]

#### For the first circle:
\[
x^2 + y^2 - 4x + 14y + 4 = 0
\]
- \(D = -4\)
- \(E = 14\)

So, the center is:
\[
h_1 = -\frac{-4}{2} = 2, \quad k_1 = -\frac{14}{2} = -7
\]
The center of the first circle is \((2, -7)\).

#### For the second circle:
\[
x^2 + y^2 + 14x + 4y + 49 = 0
\]
- \(D = 14\)
- \(E = 4\)

So, the center is:
\[
h_2 = -\frac{14}{2} = -7, \quad k_2 = -\frac{4}{2} = -2
\]
The center of the second circle is \((-7, -2)\).

### Step 2: Find the equation of the line passing through these centers.

We now have two points: \((2, -7)\) and \((-7, -2)\).

The slope \(m\) of the line passing through these two points is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - (-7)}{-7 - 2} = \frac{5}{-9} = -\frac{5}{9}
\]

Using the point-slope form of the line equation:
\[
y - y_1 = m(x - x_1)
\]
Substitute \(m = -\frac{5}{9}\) and \((x_1, y_1) = (2, -7)\):
\[
y - (-7) = -\frac{5}{9}(x - 2)
\]
Simplifying:
\[
y + 7 = -\frac{5}{9}x + \frac{10}{9}
\]
\[
y = -\frac{5}{9}x - 7 + \frac{10}{9}
\]
\[
y = -\frac{5}{9}x - \frac{53}{9}
\]

Therefore, the equation of the line passing through the centers of the two circles is:
\[
y = -\frac{5}{9}x - \frac{53}{9}
\]

Would you like a detailed explanation of any step or more information?

Here are some related questions:
1. How do you derive the center and radius of a circle from its general equation?
2. What is the significance of the slope-intercept form of a line?
3. How can you find the distance between the centers of two circles?
4. What are the different forms of equations of a line?
5. How can you determine if two circles are tangent to each other?

**Tip:** Always simplify the circle's equation to its standard form \((x - h)^2 + (y - k)^2 = r^2\) to easily find the center and radius.

Find an equation of the line containing the centers of the two circles whose equations are given below:
1. x² + y² - 4x + 14y + 4 = 0
2. x² + y² + 14x + 4y + 49 = 0

Learn how to find the equation of the line that passes through the centers of two given circles. This problem involves using the general equation of a circle, finding the slope, and applying the point-slope form of a line. Ideal for high school students studying Algebra and Coordinate Geometry.

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