Question

Calculate the radius of the circle by completing the square of the equation x2+y2−8x+16y=−44
𝑥
2
+
𝑦
2
−
8
𝑥
+
16
𝑦
=
−
44
.

Answer 1

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

Starting with the given equation:

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

### Step 1: Complete the square for \( x \)

For the \( x \) terms: \( x^2 - 8x \)

1. Take \(-8\), divide by 2 to get \(-4\), and then square it to get \(16\).
2. Rewrite the expression:

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

### Step 2: Complete the square for \( y \)

For the \( y \) terms: \( y^2 + 16y \)

1. Take \(16\), divide by 2 to get \(8\), and then square it to get \(64\).
2. Rewrite the expression:

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

### Step 3: Substitute back into the equation

Now substitute back into the original equation:

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

Combine the constants:

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

Add \(80\) to both sides:

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

### Step 4: Identify the center and radius

This equation 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.

From our equation:

- The center is \( (4, -8) \)
- The radius squared \( r^2 = 36 \)

### Step 5: Calculate the radius

Taking the square root to find the radius \( r \):

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

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