Consider the two circles
`(x-1)^(2) + (y-3)^(2) = r^(2)` and `x^(2) + y^(2) - 8x + 2y + 8 = 0`
What is the distance between the centres of the two circles ?

Options

A. 5 units
B. 6 units
C. 8 units
D. 10 units

Answer

To find the distance between the centers of the two circles given by the equations:

1. \((x-1)^{2} + (y-3)^{2} = r^{2}\)
2. \(x^{2} + y^{2} - 8x + 2y + 8 = 0\)

we will follow these steps:

### Step 1: Identify the center of the first circle
The first circle is already in standard form:
\[ (x - 1)^{2} + (y - 3)^{2} = r^{2} \]
From this equation, we can see that the center of the first circle \((x_1, y_1)\) is:
\[ (x_1, y_1) = (1, 3) \]

### Step 2: Rewrite the second circle in standard form
The second circle's equation is:
\[ x^{2} + y^{2} - 8x + 2y + 8 = 0 \]
To rewrite this in standard form, we will complete the square for both \(x\) and \(y\).

#### Completing the square for \(x\):
- Take the \(x\) terms: \(x^{2} - 8x\)
- To complete the square, we take \(-8\), halve it to get \(-4\), and square it to get \(16\).
- Thus, \(x^{2} - 8x\) becomes \((x - 4)^{2} - 16\).

#### Completing the square for \(y\):
- Take the \(y\) terms: \(y^{2} + 2y\)
- To complete the square, we take \(2\), halve it to get \(1\), and square it to get \(1\).
- Thus, \(y^{2} + 2y\) becomes \((y + 1)^{2} - 1\).

Now substituting back into the equation:
\[ (x - 4)^{2} - 16 + (y + 1)^{2} - 1 + 8 = 0 \]
This simplifies to:
\[ (x - 4)^{2} + (y + 1)^{2} - 9 = 0 \]
or
\[ (x - 4)^{2} + (y + 1)^{2} = 9 \]

### Step 3: Identify the center of the second circle
From the standard form of the second circle, we can determine its center \((x_2, y_2)\):
\[ (x_2, y_2) = (4, -1) \]

### Step 4: Calculate the distance between the two centers
We now have the centers:
- Center of the first circle: \((1, 3)\)
- Center of the second circle: \((4, -1)\)

We will use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}} \]
Substituting the values:
\[ d = \sqrt{(4 - 1)^{2} + (-1 - 3)^{2}} = \sqrt{(3)^{2} + (-4)^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Final Answer:
The distance between the centers of the two circles is \(5\) units.

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