Question
Find the center and radius of the circle represented by the equation below.
x^2 + y^2 - 14x - 4y - 47 = 0
x^2 + y^2 - 14x - 4y - 47 = 0
Answers
GPT-4o mini
To find the center and radius of the circle represented by the equation
\[
x^2 + y^2 - 14x - 4y - 47 = 0,
\]
we can start by rearranging this equation into standard form. The standard form of a circle is
\[
(x - h)^2 + (y - k)^2 = r^2,
\]
where \((h, k)\) is the center and \(r\) is the radius.
1. **Rearranging the equation:**
First, let's move \( -47 \) to the other side:
\[
x^2 + y^2 - 14x - 4y = 47.
\]
2. **Completing the square:**
- For the \(x\) terms:
\[
x^2 - 14x.
\]
To complete the square, take \(-14\) (the coefficient of \(x\)), halve it to get \(-7\), and then square it to obtain \(49\):
\[
x^2 - 14x = (x - 7)^2 - 49.
\]
- For the \(y\) terms:
\[
y^2 - 4y.
\]
To complete the square, take \(-4\), halve it to get \(-2\), and then square it to get \(4\):
\[
y^2 - 4y = (y - 2)^2 - 4.
\]
3. **Substituting back into the equation:**
Now substitute these completed squares back into the equation:
\[
(x - 7)^2 - 49 + (y - 2)^2 - 4 = 47.
\]
Simplifying this, we combine the constants on the left side:
\[
(x - 7)^2 + (y - 2)^2 - 53 = 47.
\]
Adding \(53\) to both sides gives:
\[
(x - 7)^2 + (y - 2)^2 = 100.
\]
4. **Identify the center and radius:**
Now we have the equation in standard form. From
\[
(x - 7)^2 + (y - 2)^2 = 100,
\]
we can see that:
- The center \((h, k)\) is \((7, 2)\).
- The radius \(r\) is \(\sqrt{100} = 10\).
Thus, the center and radius of the circle are:
- **Center**: \((7, 2)\)
- **Radius**: \(10\)
\[
x^2 + y^2 - 14x - 4y - 47 = 0,
\]
we can start by rearranging this equation into standard form. The standard form of a circle is
\[
(x - h)^2 + (y - k)^2 = r^2,
\]
where \((h, k)\) is the center and \(r\) is the radius.
1. **Rearranging the equation:**
First, let's move \( -47 \) to the other side:
\[
x^2 + y^2 - 14x - 4y = 47.
\]
2. **Completing the square:**
- For the \(x\) terms:
\[
x^2 - 14x.
\]
To complete the square, take \(-14\) (the coefficient of \(x\)), halve it to get \(-7\), and then square it to obtain \(49\):
\[
x^2 - 14x = (x - 7)^2 - 49.
\]
- For the \(y\) terms:
\[
y^2 - 4y.
\]
To complete the square, take \(-4\), halve it to get \(-2\), and then square it to get \(4\):
\[
y^2 - 4y = (y - 2)^2 - 4.
\]
3. **Substituting back into the equation:**
Now substitute these completed squares back into the equation:
\[
(x - 7)^2 - 49 + (y - 2)^2 - 4 = 47.
\]
Simplifying this, we combine the constants on the left side:
\[
(x - 7)^2 + (y - 2)^2 - 53 = 47.
\]
Adding \(53\) to both sides gives:
\[
(x - 7)^2 + (y - 2)^2 = 100.
\]
4. **Identify the center and radius:**
Now we have the equation in standard form. From
\[
(x - 7)^2 + (y - 2)^2 = 100,
\]
we can see that:
- The center \((h, k)\) is \((7, 2)\).
- The radius \(r\) is \(\sqrt{100} = 10\).
Thus, the center and radius of the circle are:
- **Center**: \((7, 2)\)
- **Radius**: \(10\)