Question
What is the radius of this circle? x2 + y2 −16x −10y +40 =0 .(1 point)
Responses
7
7
-7
-7
4
4
6
Responses
7
7
-7
-7
4
4
6
Answers
GPT-4o mini
To find the radius of the circle represented by the equation \(x^2 + y^2 - 16x - 10y + 40 = 0\), we first need to rewrite the equation in standard form, which is given by \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.
1. **Rewrite the equation**:
\[
x^2 - 16x + y^2 - 10y + 40 = 0
\]
2. **Group and complete the square** for \(x\) and \(y\):
- For \(x^2 - 16x\):
\[
x^2 - 16x = (x - 8)^2 - 64
\]
- For \(y^2 - 10y\):
\[
y^2 - 10y = (y - 5)^2 - 25
\]
3. **Substitute back into the equation**:
\[
(x - 8)^2 - 64 + (y - 5)^2 - 25 + 40 = 0
\]
Simplifying this:
\[
(x - 8)^2 + (y - 5)^2 - 49 = 0
\]
Thus,
\[
(x - 8)^2 + (y - 5)^2 = 49
\]
4. **Identify the radius**:
- The standard form is \((x - h)^2 + (y - k)^2 = r^2\). Here, \(r^2 = 49\).
- Taking the square root gives \(r = \sqrt{49} = 7\).
Therefore, the radius of the circle is **7**.
1. **Rewrite the equation**:
\[
x^2 - 16x + y^2 - 10y + 40 = 0
\]
2. **Group and complete the square** for \(x\) and \(y\):
- For \(x^2 - 16x\):
\[
x^2 - 16x = (x - 8)^2 - 64
\]
- For \(y^2 - 10y\):
\[
y^2 - 10y = (y - 5)^2 - 25
\]
3. **Substitute back into the equation**:
\[
(x - 8)^2 - 64 + (y - 5)^2 - 25 + 40 = 0
\]
Simplifying this:
\[
(x - 8)^2 + (y - 5)^2 - 49 = 0
\]
Thus,
\[
(x - 8)^2 + (y - 5)^2 = 49
\]
4. **Identify the radius**:
- The standard form is \((x - h)^2 + (y - k)^2 = r^2\). Here, \(r^2 = 49\).
- Taking the square root gives \(r = \sqrt{49} = 7\).
Therefore, the radius of the circle is **7**.