Question

Calculate the radius of the circle by completing the square of the equation x^2 + y^2 - 8x + 16y = -44

radius = ___

Answer 1

To find the radius of the circle from the equation \( x^2 + y^2 - 8x + 16y = -44 \), we need to complete the square for the \( x \) and \( y \) terms.

Starting with the given equation:

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

### Step 1: Rearranging the equation
We can rearrange the equation:

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

### Step 2: Completing the square for \( x \)
For the \( x \) terms \( x^2 - 8x \):
1. Take half of the coefficient of \( x \) (which is -8), square it: \((-8/2)^2 = 16\).
2. Add and subtract \( 16 \):

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

### Step 3: Completing the square for \( y \)
For the \( y \) terms \( y^2 + 16y \):
1. Take half of the coefficient of \( y \) (which is 16), square it: \((16/2)^2 = 64\).
2. Add and subtract \( 64 \):

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

### Step 4: Substitute back into the equation
Now substituting back into the equation:

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

Simplifying this:

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

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

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

### Step 5: Identifying the center and radius
This equation matches 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.

From our equation:

- Center: \((4, -8)\)
- Radius: \( r = \sqrt{36} = 6 \)

Thus, the radius of the circle is:

\[
\text{radius} = 6
\]