Question

Calculate the radius of the circle by completing the square of the equation x2+y2−8x+16y=−44
.(1 point)
radius=

Answer 1

To find the radius of the circle given by the equation \( x^2 + y^2 - 8x + 16y = -44 \) using the method of completing the square, we need to rewrite the equation in the standard form of a circle, which is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where \((h, k)\) is the center of the circle and \(r\) is the radius.

Let's start by rearranging the given equation:

\[
x^2 - 8x + y^2 + 16y = -44
\]

Now, we complete the square for the \(x\) and \(y\) terms.

### Completing the square for \(x\):
1. Take the coefficient of \(x\), which is \(-8\), divide it by \(2\) and square it:
\[
\left(-\frac{8}{2}\right)^2 = (-4)^2 = 16
\]
2. Add and subtract \(16\):
\[
x^2 - 8x + 16
\]

### Completing the square for \(y\):
1. Take the coefficient of \(y\), which is \(16\), divide it by \(2\) and square it:
\[
\left(\frac{16}{2}\right)^2 = 8^2 = 64
\]
2. Add and subtract \(64\):
\[
y^2 + 16y + 64
\]

### Now substitute back into the equation:
Putting all this back, we have:
\[
(x^2 - 8x + 16) + (y^2 + 16y + 64) = -44 + 16 + 64
\]

Which simplifies to:
\[
(x - 4)^2 + (y + 8)^2 = 36
\]

### Finding the radius:
The equation \((x - 4)^2 + (y + 8)^2 = 36\) is now in standard form, where:
- The center is \((4, -8)\)
- The radius \(r\) can be found as:
\[
r^2 = 36 \Rightarrow r = \sqrt{36} = 6
\]

Thus, the radius of the circle is:
\[
\text{radius} = 6
\]