Question
Calculate the radius of the circle by completing the square of the equation x^2 + y^2 - 14x - 4y = 11
radius = ___
radius = ___
Answers
GPT-4o mini
To find the radius of the circle from the equation \( x^2 + y^2 - 14x - 4y = 11 \), we need to complete the square for both \( x \) and \( y \).
Starting with the equation:
\[
x^2 + y^2 - 14x - 4y = 11
\]
We'll rearrange the equation to group the \( x \) and \( y \) terms:
\[
(x^2 - 14x) + (y^2 - 4y) = 11
\]
Now, we'll complete the square for the \( x \) terms. The coefficient of \( x \) is -14. Half of -14 is -7, and squaring -7 gives us 49. Thus,
\[
x^2 - 14x \rightarrow (x - 7)^2 - 49
\]
Next, we do the same for the \( y \) terms. The coefficient of \( y \) is -4. Half of -4 is -2, and squaring -2 gives us 4. Therefore,
\[
y^2 - 4y \rightarrow (y - 2)^2 - 4
\]
Now substituting these completed squares back into the equation gives us:
\[
((x - 7)^2 - 49) + ((y - 2)^2 - 4) = 11
\]
Simplifying the left side, we get:
\[
(x - 7)^2 - 49 + (y - 2)^2 - 4 = 11
\]
Combining the constants:
\[
(x - 7)^2 + (y - 2)^2 - 53 = 11
\]
Adding 53 to both sides yields:
\[
(x - 7)^2 + (y - 2)^2 = 64
\]
Now, we recognize that this is the standard form of a circle's equation:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \( (h, k) \) is the center and \( r \) is the radius. From this equation:
- The center \( (h, k) \) is \( (7, 2) \).
- The radius squared \( r^2 = 64 \).
To find the radius \( r \), we take the square root of 64:
\[
r = \sqrt{64} = 8
\]
Thus, the radius of the circle is:
\[
\text{radius} = 8
\]
So the final answer is:
\[
\text{radius} = 8
\]
Starting with the equation:
\[
x^2 + y^2 - 14x - 4y = 11
\]
We'll rearrange the equation to group the \( x \) and \( y \) terms:
\[
(x^2 - 14x) + (y^2 - 4y) = 11
\]
Now, we'll complete the square for the \( x \) terms. The coefficient of \( x \) is -14. Half of -14 is -7, and squaring -7 gives us 49. Thus,
\[
x^2 - 14x \rightarrow (x - 7)^2 - 49
\]
Next, we do the same for the \( y \) terms. The coefficient of \( y \) is -4. Half of -4 is -2, and squaring -2 gives us 4. Therefore,
\[
y^2 - 4y \rightarrow (y - 2)^2 - 4
\]
Now substituting these completed squares back into the equation gives us:
\[
((x - 7)^2 - 49) + ((y - 2)^2 - 4) = 11
\]
Simplifying the left side, we get:
\[
(x - 7)^2 - 49 + (y - 2)^2 - 4 = 11
\]
Combining the constants:
\[
(x - 7)^2 + (y - 2)^2 - 53 = 11
\]
Adding 53 to both sides yields:
\[
(x - 7)^2 + (y - 2)^2 = 64
\]
Now, we recognize that this is the standard form of a circle's equation:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \( (h, k) \) is the center and \( r \) is the radius. From this equation:
- The center \( (h, k) \) is \( (7, 2) \).
- The radius squared \( r^2 = 64 \).
To find the radius \( r \), we take the square root of 64:
\[
r = \sqrt{64} = 8
\]
Thus, the radius of the circle is:
\[
\text{radius} = 8
\]
So the final answer is:
\[
\text{radius} = 8
\]