Question
Calculate the radius of the circle by completing the square of the equation x2+y2−14x−4y=11.(1 point)radius=
Answers
GPT-4o mini
To find the radius of the circle given by the equation \( x^2 + y^2 - 14x - 4y = 11 \), we first complete the square for the \( x \) and \( y \) terms.
Starting with the equation:
\[
x^2 - 14x + y^2 - 4y = 11
\]
### Completing the square for \( x \):
1. Take the coefficient of \( x \), which is \(-14\), divide it by 2 to get \(-7\), and square it to get \(49\).
2. Add and subtract \( 49 \):
\[
x^2 - 14x + 49 - 49 + y^2 - 4y = 11
\]
\[
(x - 7)^2 - 49 + y^2 - 4y = 11
\]
### Completing the square for \( y \):
1. Take the coefficient of \( y \), which is \(-4\), divide it by 2 to get \(-2\), and square it to get \(4\).
2. Add and subtract \( 4 \):
\[
(x - 7)^2 - 49 + (y^2 - 4y + 4 - 4) = 11
\]
\[
(x - 7)^2 - 49 + (y - 2)^2 - 4 = 11
\]
### Rearranging:
Combining these terms gives:
\[
(x - 7)^2 + (y - 2)^2 - 53 = 11
\]
Now, add \( 53 \) to both sides:
\[
(x - 7)^2 + (y - 2)^2 = 64
\]
### Identifying the center and radius:
This is now in the standard form of the equation of a circle, \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
From the equation \((x - 7)^2 + (y - 2)^2 = 64\):
- The center is \((7, 2)\).
- The radius \(r\) is \(\sqrt{64} = 8\).
Thus, the radius of the circle is:
\[
\text{radius} = 8
\]
Starting with the equation:
\[
x^2 - 14x + y^2 - 4y = 11
\]
### Completing the square for \( x \):
1. Take the coefficient of \( x \), which is \(-14\), divide it by 2 to get \(-7\), and square it to get \(49\).
2. Add and subtract \( 49 \):
\[
x^2 - 14x + 49 - 49 + y^2 - 4y = 11
\]
\[
(x - 7)^2 - 49 + y^2 - 4y = 11
\]
### Completing the square for \( y \):
1. Take the coefficient of \( y \), which is \(-4\), divide it by 2 to get \(-2\), and square it to get \(4\).
2. Add and subtract \( 4 \):
\[
(x - 7)^2 - 49 + (y^2 - 4y + 4 - 4) = 11
\]
\[
(x - 7)^2 - 49 + (y - 2)^2 - 4 = 11
\]
### Rearranging:
Combining these terms gives:
\[
(x - 7)^2 + (y - 2)^2 - 53 = 11
\]
Now, add \( 53 \) to both sides:
\[
(x - 7)^2 + (y - 2)^2 = 64
\]
### Identifying the center and radius:
This is now in the standard form of the equation of a circle, \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
From the equation \((x - 7)^2 + (y - 2)^2 = 64\):
- The center is \((7, 2)\).
- The radius \(r\) is \(\sqrt{64} = 8\).
Thus, the radius of the circle is:
\[
\text{radius} = 8
\]