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 \( x^2 + y^2 - 8x + 16y = -44 \), we will first rewrite the equation by completing the square for both \( x \) and \( y \).

1. **Rearranging the equation:**

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

2. **Completing the square for \( x \):**

- Take the coefficient of \( x \) which is \(-8\), halve it to get \(-4\), and then square it to get \(16\).
- Thus, we have:
\[
x^2 - 8x = (x - 4)^2 - 16
\]

3. **Completing the square for \( y \):**

- Take the coefficient of \( y \) which is \(16\), halve it to get \(8\), and then square it to get \(64\).
- Thus, we have:
\[
y^2 + 16y = (y + 8)^2 - 64
\]

4. **Substituting these back into the equation:**

\[
(x - 4)^2 - 16 + (y + 8)^2 - 64 = -44
\]

5. **Simplifying the equation:**

\[
(x - 4)^2 + (y + 8)^2 - 80 = -44
\]

\[
(x - 4)^2 + (y + 8)^2 = -44 + 80
\]

\[
(x - 4)^2 + (y + 8)^2 = 36
\]

6. **Identifying the center and radius of the circle:**

The equation \( (x - 4)^2 + (y + 8)^2 = 36 \) is in the standard form of a circle:

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

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

Here, \( r^2 = 36 \) which gives:

\[
r = \sqrt{36} = 6
\]

Thus, the radius of the circle is:

\[
\text{radius} = 6
\]