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

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

center=(3,8), 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 the \(x\) and \(y\) terms.

1. Start with the equation:
\[
x^2 - 16x + y^2 - 6y = -24
\]

2. Complete the square for the \(x\) terms:
\[
x^2 - 16x \quad \text{can be written as} \quad (x^2 - 16x + 64 - 64) = (x - 8)^2 - 64
\]

3. Complete the square for the \(y\) terms:
\[
y^2 - 6y \quad \text{can be written as} \quad (y^2 - 6y + 9 - 9) = (y - 3)^2 - 9
\]

4. Substitute these back into the equation:
\[
(x - 8)^2 - 64 + (y - 3)^2 - 9 = -24
\]

5. Combine like terms:
\[
(x - 8)^2 + (y - 3)^2 - 73 = -24
\]
\[
(x - 8)^2 + (y - 3)^2 = 49
\]

6. Now, we can see that this is in the form of the standard equation of a circle:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center and \(r\) is the radius.

From our final equation, we have:
- The center \((h, k) = (8, 3)\)
- The radius \(r = \sqrt{49} = 7\)

Thus, the center and radius of the circle are:
- Center = \((8, 3)\), Radius = 7

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