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)`

Explanation

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 can follow these steps:

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

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

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

### Step 2: Calculate the distance between the centers
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:
- Center of the first circle: \((-a, 0)\)
- Center of the second circle: \((0, -b)\)

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

### Step 3: Conclusion
The distance between the centers of the two circles is:
\[ d = \sqrt{a^2 + b^2} \]