Question
The equation below describes a circle.
x2 - 20x + y2 + 8y = 40
What are the center and radius of the circle?
A.
The center is (10, -4) and the radius is 7.
B.
The center is (5, -2) and the radius is
√
29
.
C.
The center is (10, -4) and the radius is
√
29
.
D.
The center is (10, -4) and the radius is
√
156
.
x2 - 20x + y2 + 8y = 40
What are the center and radius of the circle?
A.
The center is (10, -4) and the radius is 7.
B.
The center is (5, -2) and the radius is
√
29
.
C.
The center is (10, -4) and the radius is
√
29
.
D.
The center is (10, -4) and the radius is
√
156
.
Answers
Answered by
GPT-4o mini
To find the center and radius of the circle from the given equation, we can rewrite the equation in the standard form of a circle, which is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
The original equation is:
\[x^2 - 20x + y^2 + 8y = 40.\]
1. **Complete the square for the x terms**:
\(x^2 - 20x\)
Take half of \(-20\), square it: \((-10)^2 = 100\).
Thus, we can rewrite the x terms:
\(x^2 - 20x = (x - 10)^2 - 100\).
2. **Complete the square for the y terms**:
\(y^2 + 8y\)
Take half of \(8\), square it: \((4)^2 = 16\).
Thus, we can rewrite the y terms:
\(y^2 + 8y = (y + 4)^2 - 16\).
3. **Substitute back into the equation**:
\((x - 10)^2 - 100 + (y + 4)^2 - 16 = 40\).
Combine the constants:
\((x - 10)^2 + (y + 4)^2 - 116 = 40\)
Therefore,
\((x - 10)^2 + (y + 4)^2 = 40 + 116\)
\((x - 10)^2 + (y + 4)^2 = 156\).
4. **Identify the center and radius**:
From the equation \((x - 10)^2 + (y + 4)^2 = 156\):
- The center \((h, k)\) is \((10, -4)\).
- The radius \(r\) is \(\sqrt{156}\).
Thus, the center of the circle is \((10, -4)\) and the radius is \(\sqrt{156}\).
The correct answer is:
**D. The center is (10, -4) and the radius is \(\sqrt{156}\).**
The original equation is:
\[x^2 - 20x + y^2 + 8y = 40.\]
1. **Complete the square for the x terms**:
\(x^2 - 20x\)
Take half of \(-20\), square it: \((-10)^2 = 100\).
Thus, we can rewrite the x terms:
\(x^2 - 20x = (x - 10)^2 - 100\).
2. **Complete the square for the y terms**:
\(y^2 + 8y\)
Take half of \(8\), square it: \((4)^2 = 16\).
Thus, we can rewrite the y terms:
\(y^2 + 8y = (y + 4)^2 - 16\).
3. **Substitute back into the equation**:
\((x - 10)^2 - 100 + (y + 4)^2 - 16 = 40\).
Combine the constants:
\((x - 10)^2 + (y + 4)^2 - 116 = 40\)
Therefore,
\((x - 10)^2 + (y + 4)^2 = 40 + 116\)
\((x - 10)^2 + (y + 4)^2 = 156\).
4. **Identify the center and radius**:
From the equation \((x - 10)^2 + (y + 4)^2 = 156\):
- The center \((h, k)\) is \((10, -4)\).
- The radius \(r\) is \(\sqrt{156}\).
Thus, the center of the circle is \((10, -4)\) and the radius is \(\sqrt{156}\).
The correct answer is:
**D. The center is (10, -4) and the radius is \(\sqrt{156}\).**