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.