Question

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

Answer 1

To find the radius of the circle, we first rewrite the given equation \( x^2 + y^2 - 8x + 16y = -44 \) by completing 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 \(2\), and square it:
\[
\left(\frac{-8}{2}\right)^2 = 16
\]
- Add and subtract \( 16 \) in the equation:
\[
x^2 - 8x + 16 + y^2 + 16y = -44 + 16
\]

3. **Completing the square for \( y \):**
- Take the coefficient of \( y \), which is \(16\), divide it by \(2\), and square it:
\[
\left(\frac{16}{2}\right)^2 = 64
\]
- Add and subtract \( 64 \) in the equation:
\[
x^2 - 8x + 16 + y^2 + 16y + 64 = -44 + 16 + 64
\]

4. **Rewriting the equation:**
Now, we can rewrite the equation as:
\[
(x - 4)^2 + (y + 8)^2 = 36
\]

5. **Identifying the circle's center and radius:**
From the standard form of a circle's equation \( (x - h)^2 + (y - k)^2 = r^2 \), we see that:
- Center \((h, k) = (4, -8)\)
- The right side \( r^2 = 36 \), so:
\[
r = \sqrt{36} = 6
\]

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