Question

A circle is represented by the equation x2+y2−16x−6y=−24 . Complete the square to discover the center and radius of the circle.(1 point)
Responses

center=(16,6), radius=8
center= left parenthesis 16 comma 6 right parenthesis , radius=8

center=(3,8), radius=7
center= left parenthesis 3 comma 8 right parenthesis , radius=7

center=(−16,−6), radius=8
center= left parenthesis negative 16 comma negative 6 right parenthesis , radius=8

center=(8,3), radius=7

Answer 1

To find the center and radius of the circle given by the equation \(x^2 + y^2 - 16x - 6y = -24\), we need to complete the square for both \(x\) and \(y\).

1. Start with the original equation:

\[
x^2 + y^2 - 16x - 6y = -24
\]

2. Rearranging it gives:

\[
x^2 - 16x + y^2 - 6y = -24
\]

3. We will complete the square for the \(x\) terms:

- Take the coefficient of \(x\) which is \(-16\), halve it to get \(-8\), and square it to get \(64\).
- Rewrite the \(x\) part:

\[
(x^2 - 16x) = (x - 8)^2 - 64
\]

4. Now complete the square for the \(y\) terms:

- Take the coefficient of \(y\) which is \(-6\), halve it to get \(-3\), and square it to get \(9\).
- Rewrite the \(y\) part:

\[
(y^2 - 6y) = (y - 3)^2 - 9
\]

5. Substitute these back into the equation:

\[
((x - 8)^2 - 64) + ((y - 3)^2 - 9) = -24
\]

Simplifying gives:

\[
(x - 8)^2 + (y - 3)^2 - 73 = -24
\]

\[
(x - 8)^2 + (y - 3)^2 = 49
\]

6. Now we can find the center and radius:

- The center \((h, k)\) is \((8, 3)\).
- The radius \(r\) is \(\sqrt{49} = 7\).

Therefore, the center is \((8, 3)\) and the radius is \(7\).

The correct response is:
**center=(8,3), radius=7**