Calculate the center and radius of the circle by completing the square of the equation x2+y2+12x−18y+17=0 .(1 point)

Question

Responses

center=(6,−9), radius=10
center= left parenthesis 6 comma negative 9 right parenthesis , radius=10

center=(−6,9), radius=10
center= left parenthesis negative 6 comma 9 right parenthesis , radius=10

center=(12,−18), radius=20
center= left parenthesis 12 comma negative 18 right parenthesis , radius=20

center=(−12,18), radius=20

Answer 1

To find the center and radius of the circle given by the equation \( x^2 + y^2 + 12x - 18y + 17 = 0 \), we will complete the square for both the \( x \) and \( y \) terms.

1. Rearrange the equation: \[ x^2 + 12x + y^2 - 18y + 17 = 0 \] \[ x^2 + 12x + y^2 - 18y = -17 \]

2. Complete the square for the \( x \) terms: The terms involving \( x \) are \( x^2 + 12x \). To complete the square, take half of 12, square it, and add/subtract it: \[ x^2 + 12x = (x + 6)^2 - 36 \]

3. Complete the square for the \( y \) terms: The terms involving \( y \) are \( y^2 - 18y \). To complete the square, take half of -18, square it, and add/subtract it: \[ y^2 - 18y = (y - 9)^2 - 81 \]

4. Substitute back into the equation: \[ (x + 6)^2 - 36 + (y - 9)^2 - 81 = -17 \]

   Combine constant terms: \[ (x + 6)^2 + (y - 9)^2 - 117 = -17 \] \[ (x + 6)^2 + (y - 9)^2 = 100 \]

5. Identify the center and radius: The equation now is in the standard form of a circle: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \((h, k)\) is the center and \(r\) is the radius. Thus:

   • Center: \((-6, 9)\)
   • Radius: \(\sqrt{100} = 10\)

Therefore, the correct answer is: center = (-6, 9), radius = 10.