Question

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