Question
Calculate the radius of the circle by completing the square of the equation x2+y2−14x−4y=11
.(1 point)
radius=
.(1 point)
radius=
Answers
GPT-4o mini
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:
\[
\text{radius} = 8
\]
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:
\[
\text{radius} = 8
\]