Question

Calculate the center and radius of the circle by completing the square of the equation  x2+y2+12x−18y+17=0 .(1 point)Responsescenter=(−12,18), radius=20  center= left parenthesis negative 12 comma 18 right parenthesis , radius=20  center=(−6,9), radius=10center= left parenthesis negative 6 comma 9 right parenthesis , radius=10center=(6,−9), radius=10  center= left parenthesis 6 comma negative 9 right parenthesis , radius=10  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 the \( x \) and \( y \) terms.

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

2. Move the constant term to the other side:
\[
x^2 + y^2 + 12x - 18y = -17
\]

3. Complete the square for the \( x \) terms:
- Take the coefficient of \( x \) (which is 12), divide it by 2 to get 6, and square it to get \( 36 \).
- Rewrite the \( x \) expression:
\[
x^2 + 12x = (x + 6)^2 - 36
\]

4. Complete the square for the \( y \) terms:
- Take the coefficient of \( y \) (which is -18), divide it by 2 to get -9, and square it to get \( 81 \).
- Rewrite the \( y \) expression:
\[
y^2 - 18y = (y - 9)^2 - 81
\]

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

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

Move \( -117 \) to the other side:
\[
(x + 6)^2 + (y - 9)^2 = -17 + 117
\]
\[
(x + 6)^2 + (y - 9)^2 = 100
\]

7. Identify the standard form of the circle:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Here, \( h = -6 \), \( k = 9 \), and \( r^2 = 100 \).

8. Calculate the radius:
\[
r = \sqrt{100} = 10
\]

Thus, the center of the circle is \( (-6, 9) \) and the radius is \( 10 \).

**Final Answer:**
Center: \( (-6, 9) \), Radius: \( 10 \)