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

Question

Responses

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

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

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

center=(2,−8), radius=25

Answer 1

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

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

2. Complete the square for the \( x \) terms:

   • Take the coefficient of \( x \) (which is -8), halve it to get -4, and square it to get 16.
   • Thus, \( x^2 - 8x \) becomes \( (x - 4)^2 - 16 \).

3. Complete the square for the \( y \) terms:

   • Take the coefficient of \( y \) (which is 2), halve it to get 1, and square it to get 1.
   • Thus, \( y^2 + 2y \) becomes \( (y + 1)^2 - 1 \).

4. Substitute these completed squares back into the equation: \[ (x - 4)^2 - 16 + (y + 1)^2 - 1 = 8 \]

5. Simplify the equation: \[ (x - 4)^2 + (y + 1)^2 - 17 = 8 \] \[ (x - 4)^2 + (y + 1)^2 = 25 \]

Now we can identify the center and radius:

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

Therefore, the center and radius of the circle are:

• Center = (4, -1), Radius = 5

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