Consider the circles `x^(2) + y^(2) + 2ax + c = 0` and `x^(2) + y^(2) + 2by + c = 0`
What is the distance between the centres of the two circles ?

Options

A. `sqrt(a^(2) + b^(2))`
B. `a^(2) + b^(2)`
C. `a + b`
D. `2 (a + b)`

Answer

To find the distance between the centers of the two circles given by the equations \(x^2 + y^2 + 2ax + c = 0\) and \(x^2 + y^2 + 2by + c = 0\), we will follow these steps:

### Step 1: Identify the centers of the circles
The general form of a circle is given by the equation:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From this, we can identify the center of the circle as \((-g, -f)\).

For the first circle \(x^2 + y^2 + 2ax + c = 0\):
- Here, \(g = a\) and \(f = 0\).
- Therefore, the center \(C_1\) is \((-a, 0)\).

For the second circle \(x^2 + y^2 + 2by + c = 0\):
- Here, \(g = 0\) and \(f = b\).
- Therefore, the center \(C_2\) is \((0, -b)\).

### Step 2: Use the distance formula
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of the centers \(C_1(-a, 0)\) and \(C_2(0, -b)\):
- \(x_1 = -a\), \(y_1 = 0\)
- \(x_2 = 0\), \(y_2 = -b\)

Now substituting these values into the distance formula:
\[ d = \sqrt{(0 - (-a))^2 + (-b - 0)^2} \]
\[ d = \sqrt{(a)^2 + (-b)^2} \]
\[ d = \sqrt{a^2 + b^2} \]

### Conclusion
The distance between the centers of the two circles is:
\[ d = \sqrt{a^2 + b^2} \]