Find the distance between the centres of the circles `x^(2)+y^(2)+8x+10y-2=0andx^(2)+y^(2)+2x+2y-1=0`.

Answer

To find the distance between the centers of the given circles, we will follow these steps:

### Step 1: Write the equations of the circles
The equations of the circles are:
1. \( x^2 + y^2 + 8x + 10y - 2 = 0 \) (Circle 1)
2. \( x^2 + y^2 + 2x + 2y - 1 = 0 \) (Circle 2)

### Step 2: Compare with the general form of the circle
The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

For Circle 1:
- \( 2g = 8 \) ⇒ \( g = 4 \)
- \( 2f = 10 \) ⇒ \( f = 5 \)

For Circle 2:
- \( 2g = 2 \) ⇒ \( g = 1 \)
- \( 2f = 2 \) ⇒ \( f = 1 \)

### Step 3: Find the centers of the circles
The center of a circle is given by the coordinates \( (-g, -f) \).

For Circle 1:
- Center \( C_1 = (-4, -5) \)

For Circle 2:
- Center \( C_2 = (-1, -1) \)

### Step 4: Use the distance formula
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of the centers:
- \( C_1 = (-4, -5) \)
- \( C_2 = (-1, -1) \)

We calculate:
\[ d = \sqrt{((-1) - (-4))^2 + ((-1) - (-5))^2} \]
\[ d = \sqrt{(-1 + 4)^2 + (-1 + 5)^2} \]
\[ d = \sqrt{(3)^2 + (4)^2} \]
\[ d = \sqrt{9 + 16} \]
\[ d = \sqrt{25} \]
\[ d = 5 \]

### Conclusion
The distance between the centers of the circles is \( 5 \) units.

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