Question

find the center and the radius of the following circle.
x² + 16x + y² + 12y = 0

the center is (-8, -6).
(type an ordered pair.)
the radius is □.
(simplify your answer.)

Explanation:

Step1: Complete the square for x-terms

For \(x^2 + 16x\), take half of 16 (which is 8), square it (\(8^2 = 64\)). So we add and subtract 64.
For \(y^2 + 12y\), take half of 12 (which is 6), square it (\(6^2 = 36\)). So we add and subtract 36.
The equation becomes:
\(x^2 + 16x + 64 + y^2 + 12y + 36 = 64 + 36\)

Step2: Rewrite in standard form

The left side factors into \((x + 8)^2 + (y + 6)^2\), and the right side is \(100\).
The standard form of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
Here, \((x + 8)^2 + (y + 6)^2 = 10^2\), so \(r^2 = 100\).

Step3: Find the radius

Take the square root of \(r^2\) to get \(r\). Since radius is positive, \(r = \sqrt{100} = 10\).

Answer:

10