Question
A circle drawn on the coordinate plane has a diameter with endpoints given by points A(-1,2) and B(3, 2). Which equation describes the circle?
A.
x2 + y2 = 4
B.
x2 - y2 = 4
C.
(x + 1)2 + (y + 2)2 = 4
D.
(x - 2)2 + (y - 2)2 = 4
A.
x2 + y2 = 4
B.
x2 - y2 = 4
C.
(x + 1)2 + (y + 2)2 = 4
D.
(x - 2)2 + (y - 2)2 = 4
Answers
Answered by
GPT-4o mini
To find the equation of the circle with a diameter defined by points A(-1, 2) and B(3, 2), we can follow these steps:
1. **Find the center of the circle**: The center of the circle is the midpoint of the diameter. The midpoint (M) can be calculated using the formula:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Applying the coordinates of points A and B:
\[
M = \left( \frac{-1 + 3}{2}, \frac{2 + 2}{2} \right) = \left( \frac{2}{2}, 2 \right) = (1, 2)
\]
2. **Find the radius of the circle**: The radius is half the distance of the diameter. The distance between points A and B can be calculated as:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
For points A(-1, 2) and B(3, 2):
\[
d = \sqrt{(3 - (-1))^2 + (2 - 2)^2} = \sqrt{(3 + 1)^2 + 0} = \sqrt{4^2} = 4
\]
Therefore, the radius r is:
\[
r = \frac{d}{2} = \frac{4}{2} = 2
\]
3. **Write the equation of the circle**: The general equation for a circle centered at (h, k) with a radius r is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Here, the center (h, k) is (1, 2) and the radius r is 2. Plugging these values into the equation gives:
\[
(x - 1)^2 + (y - 2)^2 = 2^2
\]
Simplifying this results in:
\[
(x - 1)^2 + (y - 2)^2 = 4
\]
Now, we check the provided options:
- A. \(x^2 + y^2 = 4\)
- B. \(x^2 - y^2 = 4\)
- C. \((x + 1)^2 + (y + 2)^2 = 4\)
- D. \((x - 2)^2 + (y - 2)^2 = 4\)
None of these match perfectly with \((x - 1)^2 + (y - 2)^2 = 4\). However, since it is not explicitly present in any of the options, the correct circle equation is \((x - 1)^2 + (y - 2)^2 = 4\).
This suggests that the options might be incorrect, or it's worth noting that none of the options correctly represent the circle described by points A and B.
1. **Find the center of the circle**: The center of the circle is the midpoint of the diameter. The midpoint (M) can be calculated using the formula:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Applying the coordinates of points A and B:
\[
M = \left( \frac{-1 + 3}{2}, \frac{2 + 2}{2} \right) = \left( \frac{2}{2}, 2 \right) = (1, 2)
\]
2. **Find the radius of the circle**: The radius is half the distance of the diameter. The distance between points A and B can be calculated as:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
For points A(-1, 2) and B(3, 2):
\[
d = \sqrt{(3 - (-1))^2 + (2 - 2)^2} = \sqrt{(3 + 1)^2 + 0} = \sqrt{4^2} = 4
\]
Therefore, the radius r is:
\[
r = \frac{d}{2} = \frac{4}{2} = 2
\]
3. **Write the equation of the circle**: The general equation for a circle centered at (h, k) with a radius r is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Here, the center (h, k) is (1, 2) and the radius r is 2. Plugging these values into the equation gives:
\[
(x - 1)^2 + (y - 2)^2 = 2^2
\]
Simplifying this results in:
\[
(x - 1)^2 + (y - 2)^2 = 4
\]
Now, we check the provided options:
- A. \(x^2 + y^2 = 4\)
- B. \(x^2 - y^2 = 4\)
- C. \((x + 1)^2 + (y + 2)^2 = 4\)
- D. \((x - 2)^2 + (y - 2)^2 = 4\)
None of these match perfectly with \((x - 1)^2 + (y - 2)^2 = 4\). However, since it is not explicitly present in any of the options, the correct circle equation is \((x - 1)^2 + (y - 2)^2 = 4\).
This suggests that the options might be incorrect, or it's worth noting that none of the options correctly represent the circle described by points A and B.