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 represented by the given equation \(x^2 + y^2 - 8x + 16y = -44\), we will complete the square for both the \(x\) and \(y\) terms.

1. **Rearranging the equation:**
\[
x^2 - 8x + y^2 + 16y = -44
\]

2. **Completing the square for \(x\):**
- Take the coefficient of \(x\), which is \(-8\), divide it by 2 to get \(-4\), and then square it: \((-4)^2 = 16\).
- Add and subtract \(16\) inside the equation.
\[
(x^2 - 8x + 16) + y^2 + 16y = -44 + 16
\]
This simplifies to:
\[
(x - 4)^2 + y^2 + 16y = -28
\]

3. **Completing the square for \(y\):**
- Take the coefficient of \(y\), which is \(16\), divide it by 2 to get \(8\), and square it: \(8^2 = 64\).
- Add and subtract \(64\) inside the equation.
\[
(x - 4)^2 + (y^2 + 16y + 64) = -28 + 64
\]
This simplifies to:
\[
(x - 4)^2 + (y + 8)^2 = 36
\]

4. **Identify the center and radius:**
From the standard form of a circle \((x - h)^2 + (y - k)^2 = r^2\), we can see:
- The center \((h, k)\) is \((4, -8)\).
- The right side \(36\) represents \(r^2\).

5. **Calculating the radius:**
\[
r = \sqrt{36} = 6
\]

Thus, the radius of the circle is \(\boxed{6}\).