Question

Calculate the radius of the circle by completing the square of the equation x2+y2−14x−4y=11
.(1 point)
radius=

Answer 1

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

1. **Rearrange the equation:**

\[
x^2 - 14x + y^2 - 4y = 11
\]

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

For the term \( x^2 - 14x \):
- Take half of the coefficient of \( x \), which is \(-14\), giving \(-7\).
- Square it: \((-7)^2 = 49\).

So, we rewrite \( x^2 - 14x \) as:

\[
(x - 7)^2 - 49
\]

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

For the term \( y^2 - 4y \):
- Take half of the coefficient of \( y \), which is \(-4\), giving \(-2\).
- Square it: \((-2)^2 = 4\).

So, we rewrite \( y^2 - 4y \) as:

\[
(y - 2)^2 - 4
\]

4. **Substitute back into the equation:**

Replacing \( x^2 - 14x \) and \( y^2 - 4y \) in the equation we have:

\[
(x - 7)^2 - 49 + (y - 2)^2 - 4 = 11
\]

5. **Combine constant terms:**

\[
(x - 7)^2 + (y - 2)^2 - 53 = 11
\]

Adding \( 53 \) to both sides gives:

\[
(x - 7)^2 + (y - 2)^2 = 64
\]

6. **Identify the center and radius:**

The equation \((x - 7)^2 + (y - 2)^2 = 64\) is in standard form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \( r \) is the radius.

Here, \( r^2 = 64 \), so:

\[
r = \sqrt{64} = 8
\]

Thus, the radius of the circle is:

\[
\text{radius} = 8
\]