Question

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

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

center=(2,−8)
, 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 the \( x \) and \( y \) terms.

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

2. **Completing the square for \( x \)**:
- Take the coefficient of \( x \) (which is -8), halve it to get -4, and then square it to get 16.
- Add and subtract 16:
\[
x^2 - 8x + 16 - 16
\]
This gives us:
\[
(x - 4)^2 - 16
\]

3. **Completing the square for \( y \)**:
- Take the coefficient of \( y \) (which is 2), halve it to get 1, and then square it to get 1.
- Add and subtract 1:
\[
y^2 + 2y + 1 - 1
\]
This gives us:
\[
(y + 1)^2 - 1
\]

4. **Putting it all together**:
Substituting back into the equation:
\[
(x - 4)^2 - 16 + (y + 1)^2 - 1 = 8
\]
Solving for the constant terms:
\[
(x - 4)^2 + (y + 1)^2 - 17 = 8
\]
\[
(x - 4)^2 + (y + 1)^2 = 25
\]

5. **Identifying the center and radius**:
The equation \( (x - 4)^2 + (y + 1)^2 = 25 \) tells us that the center of the circle is \( (4, -1) \), and the radius is \( \sqrt{25} = 5 \).

Therefore, the correct answer is:

**center=(4,−1), radius=5**