To determine the coordinates of the center and the radius of the circle from the given equation:
[tex]\[
x^2 + y^2 - 4x - 10y + 20 = 0
\][/tex]
we need to put the equation into the standard form of a circle, which is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
1. Rearrange the Equation:
We start with the given equation:
[tex]\[
x^2 + y^2 - 4x - 10y + 20 = 0
\][/tex]
2. Complete the Square for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] Terms:
To complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms, we separate and rearrange the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
[tex]\[
x^2 - 4x + y^2 - 10y = -20
\][/tex]
For [tex]\(x\)[/tex] terms:
The expression [tex]\(x^2 - 4x\)[/tex] can be rewritten by completing the square.
[tex]\[
x^2 - 4x \quad \text{becomes} \quad (x - 2)^2 - 4
\][/tex]
For [tex]\(y\)[/tex] terms:
The expression [tex]\(y^2 - 10y\)[/tex] can be rewritten by completing the square.
[tex]\[
y^2 - 10y \quad \text{becomes} \quad (y - 5)^2 - 25
\][/tex]
3. Rewrite the Equation:
Substitute the completed squares back into the equation:
[tex]\[
(x - 2)^2 - 4 + (y - 5)^2 - 25 = -20
\][/tex]
Combine like terms:
[tex]\[
(x - 2)^2 + (y - 5)^2 - 29 = -20
\][/tex]
Simplify the equation:
[tex]\[
(x - 2)^2 + (y - 5)^2 - 29 = -20
\][/tex]
[tex]\[
(x - 2)^2 + (y - 5)^2 = 9
\][/tex]
4. Identify the Center and Radius:
The equation [tex]\((x - 2)^2 + (y - 5)^2 = 9\)[/tex] matches the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where:
- [tex]\(h = 2\)[/tex]
- [tex]\(k = 5\)[/tex]
- [tex]\(r^2 = 9\)[/tex], so [tex]\(r = \sqrt{9} = 3\)[/tex]
Therefore, the coordinates of the center of the circle are [tex]\((2, 5)\)[/tex] and the radius is [tex]\(3\)[/tex] units.
Thus, the correct answer is:
C. center: [tex]\((2, 5)\)[/tex]
radius: 3 units
