To determine the center and radius of the circle defined by the given equation [tex]\( x^2 + y^2 - 4x - 10y + 20 = 0 \)[/tex], we need to rewrite this equation in the standard form of a circle's equation, 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 and group the terms:
First, we rearrange the equation to group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
[tex]\[
x^2 - 4x + y^2 - 10y = -20
\][/tex]
2. Complete the square for the [tex]\(x\)[/tex] terms:
To complete the square for [tex]\(x^2 - 4x\)[/tex]:
[tex]\[
x^2 - 4x
\][/tex]
We take half the coefficient of [tex]\(x\)[/tex] (which is [tex]\( -4 \)[/tex]), square it (giving us [tex]\( 4 \)[/tex]), and add and subtract this square within the equation:
[tex]\[
x^2 - 4x + 4 - 4 = (x - 2)^2 - 4
\][/tex]
3. Complete the square for the [tex]\(y\)[/tex] terms:
Similarly for [tex]\(y^2 - 10y\)[/tex]:
[tex]\[
y^2 - 10y
\][/tex]
We take half the coefficient of [tex]\(y\)[/tex] (which is [tex]\(-10\)[/tex]), square it (giving us [tex]\(25\)[/tex]), and add and subtract this square within the equation:
[tex]\[
y^2 - 10y + 25 - 25 = (y - 5)^2 - 25
\][/tex]
4. Substitute the completed squares back into the equation:
Substitute [tex]\((x - 2)^2 - 4\)[/tex] and [tex]\((y - 5)^2 - 25\)[/tex] into the equation:
[tex]\[
(x - 2)^2 - 4 + (y - 5)^2 - 25 = -20
\][/tex]
5. Combine constants and simplify:
Combine the constants remaining on the left side:
[tex]\[
(x - 2)^2 + (y - 5)^2 - 29 = -20
\][/tex]
Rearrange to isolate the completed squares:
[tex]\[
(x - 2)^2 + (y - 5)^2 = -20 + 29
\][/tex]
Simplifying further:
[tex]\[
(x - 2)^2 + (y - 5)^2 = 9
\][/tex]
6. Identify the center and radius:
The equation [tex]\((x - 2)^2 + (y - 5)^2 = 9\)[/tex] is now in the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where:
- Center [tex]\((h, k) = (2, 5)\)[/tex]
- Radius [tex]\(r = \sqrt{9} = 3\)[/tex]
Therefore, the center of the circle is at [tex]\((2, 5)\)[/tex] and the radius is 3 units.
The correct choice is:
B. center: [tex]\((2, 5)\)[/tex], radius: 3 units
