Question

Calculate the radius of the circle by completing the square of the equation x2+y2−14x−4y=11
.(1 point)
radius=

Answer 1

To find the radius of the circle from the equation \( x^2 + y^2 - 14x - 4y = 11 \), we will first rewrite the equation by completing the square for both the \( x \) and \( y \) terms.

1. Start with the original equation:

\[
x^2 + y^2 - 14x - 4y = 11
\]

2. Rearrange the equation:

\[
x^2 - 14x + y^2 - 4y = 11
\]

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

- Take the coefficient of \( x \) (which is -14), divide by 2 to get -7, and then square it to get 49.
- Add and subtract 49:

\[
(x^2 - 14x + 49) - 49
\]

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

- Take the coefficient of \( y \) (which is -4), divide by 2 to get -2, and then square it to get 4.
- Add and subtract 4:

\[
(y^2 - 4y + 4) - 4
\]

5. Now rewrite the equation with both squares completed:

\[
(x - 7)^2 - 49 + (y - 2)^2 - 4 = 11
\]

6. Simplify this equation:

\[
(x - 7)^2 + (y - 2)^2 - 53 = 11
\]

\[
(x - 7)^2 + (y - 2)^2 = 64
\]

7. The equation \((x - 7)^2 + (y - 2)^2 = 64\) 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.

8. From our equation, we see:

\[
r^2 = 64
\]

Taking the square root to find the radius:

\[
r = \sqrt{64} = 8
\]

Therefore, the radius of the circle is \( \boxed{8} \).