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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 4 comma negative 1 right parenthesis , radius=5

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

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

center=(−4,1), radius=5

1 answer

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.

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

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

    • Take \( x^2 - 8x \):
      • The term needed to complete the square is \( \left(\frac{-8}{2}\right)^2 = 16 \).
      • So, \( x^2 - 8x = (x - 4)^2 - 16 \).

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

    • Take \( y^2 + 2y \):
      • The term needed to complete the square is \( \left(\frac{2}{2}\right)^2 = 1 \).
      • So, \( y^2 + 2y = (y + 1)^2 - 1 \).

  4. Now substituting back into the equation: \[ (x - 4)^2 - 16 + (y + 1)^2 - 1 = 8 \]

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

  6. From this, we can identify the center and radius:

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

Therefore, the center and radius of the circle are: center=(4,−1), radius=5.