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 \(x\):
- Take the coefficient of \(x\), which is \(-8\), divide it by \(2\) and square it: \[ \left(-\frac{8}{2}\right)^2 = (-4)^2 = 16 \]
- Add and subtract \(16\): \[ x^2 - 8x + 16 \]
Completing the square for \(y\):
- Take the coefficient of \(y\), which is \(16\), divide it by \(2\) and square it: \[ \left(\frac{16}{2}\right)^2 = 8^2 = 64 \]
- 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: \[ \text{radius} = 6 \]