Question

Using the equation x2+y2−8x+2y=8
, calculate the center and radius of the circle by completing the square.(1 point)
Responses

center=(−4,1)
, radius=5
center= left parenthesis negative 4 comma 1 right parenthesis , radius=5

center=(2,−8)
, radius=25
center= left parenthesis 2 comma negative 8 right parenthesis , radius=25

center=(4,−1)
, radius=5
center= left parenthesis 4 comma negative 1 right parenthesis , radius=5

center=(8,−2)
, radius=25

Answer 1

To find the center and radius of the circle given by the equation \( x^2 + y^2 - 8x + 2y = 8 \), we can complete the square for both \( x \) and \( y \).

1. Start with the equation:
\[
x^2 + y^2 - 8x + 2y = 8
\]

2. Rearrange the equation:
\[
x^2 - 8x + y^2 + 2y = 8
\]

3. Complete the square for \( x \):
- Take the coefficient of \( x \) (-8), halve it to get -4, and square it to get 16.
- Add and subtract 16:
\[
(x^2 - 8x + 16) - 16
\]

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

5. Substitute these back into the equation:
\[
(x^2 - 8x + 16) + (y^2 + 2y + 1) = 8 + 16 + 1
\]
Which simplifies to:
\[
(x - 4)^2 + (y + 1)^2 = 25
\]

From this equation, we can see that it is in the standard circle form \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center, and \( r \) is the radius.

- The center \( (h, k) \) is \( (4, -1) \).
- The radius \( r \) is \( \sqrt{25} = 5 \).

Thus, the center and radius of the circle are:
- Center: \( (4, -1) \)
- Radius: \( 5 \)

The correct response is:
**center=(4,−1), radius=5**