The diameter of a circle has endpoints at (-2,3) and (6,3). What is an equation of the circle?
the midpoiint of the diameter (2,3) is the center of the circle.
So the equation is (x-2)^2 + (y-3)^2 = r^2
r is half the length of the diameter, so ...
A(-2, 3), C(x, y), B(6, 3).
x+2 = 1/2(6+2)
X = 2.
y-3 = 1/2(3-3)
Y = 3.
Eq: (x+2)^2+(y-3)^2 = r^2.
To find the equation of a circle with the given endpoints of the diameter, we need to find the center and radius using the midpoint formula.
Step 1: Find the coordinates of the midpoint.
The midpoint of a line segment with endpoints A(x1, y1) and B(x2, y2) is given by:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Using the given endpoints (-2, 3) and (6, 3):
Midpoint = ((-2 + 6)/2, (3 + 3)/2)
= (4/2, 6/2)
= (2, 3)
So, the midpoint of the diameter is (2, 3).
Step 2: Find the radius.
The radius is the distance from the center to any endpoint of the diameter. We can use the distance formula to find it.
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of the midpoint (2, 3) and one endpoint (-2, 3):
Distance = sqrt((2 - (-2))^2 + (3 - 3)^2)
= sqrt((2 + 2)^2 + (0)^2)
= sqrt(4^2 + 0^2)
= sqrt(16)
= 4
So, the radius of the circle is 4.
Step 3: Write the equation of the circle.
The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Using the center (2, 3) and radius 4:
(x - 2)^2 + (y - 3)^2 = 4^2
(x - 2)^2 + (y - 3)^2 = 16
Therefore, the equation of the circle is (x - 2)^2 + (y - 3)^2 = 16.
To find the equation of a circle given its endpoints, you can follow these steps:
1. Find the center of the circle:
- The center of a circle is located at the midpoint of its diameter.
- To find the midpoint, use the formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
- Plug in the coordinates of the endpoints (-2,3) and (6,3) into the formula to find the midpoint.
Midpoint = ((-2 + 6) / 2, (3 + 3) / 2)
= (4 / 2, 6 / 2)
= (2, 3)
Therefore, the center of the circle is (2, 3).
2. Find the radius of the circle:
- The radius of a circle is equal to half the length of its diameter.
- The distance formula can be used to find the length of the diameter.
- The formula for distance between two points ((x1, y1) and (x2, y2)) is:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
- Calculate the distance between the two endpoints.
Distance = sqrt((6 - (-2))^2 + (3 - 3)^2)
= sqrt((8)^2 + (0)^2)
= sqrt(64 + 0)
= sqrt(64)
= 8
Therefore, the diameter of the circle is 8 units, and the radius is half of that, 4 units.
3. Write the equation of the circle:
- The standard equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represent the center of the circle, and r represents the radius.
Plugging in the values into the equation, we get:
(x - 2)^2 + (y - 3)^2 = 4^2
Simplifying the equation, the final answer is:
(x - 2)^2 + (y - 3)^2 = 16
Therefore, the equation of the circle is (x - 2)^2 + (y - 3)^2 = 16.