Question

Regina is trying to sketch a circle on graph paper with the equation  x2+y2−2x+22y+1=0 . Calculate where she would need to graph the center and radius of the circle by completing the square.(1 point)Responsescenter=(−22,2), radius=41center= left parenthesis negative 22 comma 2 right parenthesis , radius=41center=(−2,22), radius=41center= left parenthesis negative 2 comma 22 right parenthesis , radius=41center=(1,−11), radius=11center= left parenthesis 1 comma negative 11 right parenthesis , radius=11center=(−1,11), radius=11

Answer 1

To find the center and radius of the circle given by the equation \(x^2 + y^2 - 2x + 22y + 1 = 0\), we need to complete the square.

1. Rearrange the equation:
\[
x^2 - 2x + y^2 + 22y + 1 = 0
\]
\[
x^2 - 2x + y^2 + 22y = -1
\]

2. Complete the square for the \(x\) terms:
- Take the coefficient of \(x\) (which is -2), halve it to get -1, and square it to get 1.
- Add and subtract 1:
\[
(x^2 - 2x + 1) - 1
\]

3. Complete the square for the \(y\) terms:
- Take the coefficient of \(y\) (which is 22), halve it to get 11, and square it to get 121.
- Add and subtract 121:
\[
(y^2 + 22y + 121) - 121
\]

4. Now substitute these completed squares back into the equation:
\[
(x - 1)^2 - 1 + (y + 11)^2 - 121 = -1
\]
Combine the constants:
\[
(x - 1)^2 + (y + 11)^2 - 122 = -1
\]
\[
(x - 1)^2 + (y + 11)^2 = 121
\]

5. Now we can identify the center and the radius:
- The center is \((h, k) = (1, -11)\)
- The radius \(r = \sqrt{121} = 11\)

Thus, the center and radius of the circle are:
\[
\text{center} = (1, -11), \quad \text{radius} = 11
\]

The correct response is:
**center=(1,−11), radius=11**.