Question

A circle is represented by the equation x^2 + y^2 - 16x - 6y = -24. Complete the square to discover the center and radius of the circle.

a. center = (16,6), radius = 8
b. center = (-16, -6), radius = 8
c. center = (8,3), radius = 7
d. center = (3,8), radius = 7

Answer 1

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

First, we rearrange the equation so that we can handle the \(x\) and \(y\) terms separately:

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

Now, let's complete the square for the \(x\) terms and \(y\) terms.

**For the \(x\) terms:**

1. Take the coefficient of \(x\), which is \(-16\), divide by 2 to get \(-8\), and then square it to get \(64\).
2. Add and subtract \(64\) in the equation.

**For the \(y\) terms:**

1. Take the coefficient of \(y\), which is \(-6\), divide by 2 to get \(-3\), and then square it to get \(9\).
2. Add and subtract \(9\) in the equation.

So we can rewrite the equation as:

\[
(x^2 - 16x + 64) + (y^2 - 6y + 9) = -24 + 64 + 9
\]

This simplifies to:

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

Now we can identify the center and the radius from the standard form of the circle's equation, which is \((x - h)^2 + (y - k)^2 = r^2\):

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

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

Therefore, the correct answer is:

**c. center = (8, 3), radius = 7**.