Question

calculate the radius of the circle by completing the square of the equation
x2 + y2 - 14x - 4y = 11

Answer 1

To find the radius of the circle given the equation \(x^2 + y^2 - 14x - 4y = 11\), we will complete the square for both the \(x\) and \(y\) terms.

**Step 1: Rearranging the equation**
Start with the given equation:
\[
x^2 + y^2 - 14x - 4y = 11
\]
We can rearrange it to group the \(x\) terms and the \(y\) terms together:
\[
x^2 - 14x + y^2 - 4y = 11
\]

**Step 2: Completing the square for \(x\)**
To complete the square for the \(x\) terms \(x^2 - 14x\):
1. Take half of the coefficient of \(x\) (which is \(-14\)), square it: \((-14/2)^2 = 49\).
2. Add and subtract \(49\) inside the equation:
\[
(x^2 - 14x + 49) - 49
\]
This becomes:
\[
(x - 7)^2 - 49
\]

**Step 3: Completing the square for \(y\)**
Next, complete the square for the \(y\) terms \(y^2 - 4y\):
1. Take half of the coefficient of \(y\) (which is \(-4\)), square it: \((-4/2)^2 = 4\).
2. Add and subtract \(4\):
\[
(y^2 - 4y + 4) - 4
\]
This becomes:
\[
(y - 2)^2 - 4
\]

**Step 4: Substitute back into the equation**
Now substitute back these completed squares into the equation:
\[
((x - 7)^2 - 49) + ((y - 2)^2 - 4) = 11
\]
This simplifies to:
\[
(x - 7)^2 + (y - 2)^2 - 53 = 11
\]
Adding \(53\) to both sides gives:
\[
(x - 7)^2 + (y - 2)^2 = 64
\]

**Step 5: Identify the center and radius**
Now, we can identify the standard form of the circle's equation, which is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

From our equation:
- The center \((h, k) = (7, 2)\)
- The right side \(64\) means \(r^2 = 64\)

Thus, the radius \(r\) is:
\[
r = \sqrt{64} = 8
\]

Therefore, the radius of the circle is \( \boxed{8} \).