given a circle with (5,1) and (3,-1) as the endpoint of the diameter
centre is the midpoint of the diameter, so
(4,0)
equation: (x-4)^2 + y^2 = r^2
plug in (5,1) to find r^2
all done!
To find the equation of a circle given the endpoints of a diameter, you can use the midpoint formula to find the center of the circle and then use the distance formula to find the radius.
Step 1: Finding the center of the circle
The center of the circle is simply the midpoint of the diameter, which can be found by averaging the x-coordinates and the y-coordinates of the two endpoints.
Average of x-coordinates: (5 + 3) / 2 = 4
Average of y-coordinates: (1 + (-1)) / 2 = 0
Therefore, the center of the circle is at (4, 0).
Step 2: Finding the radius of the circle
The radius of the circle is half the length of the diameter, which can be found using the distance formula.
Distance Formula: √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates (5, 1) and (3, -1) as (x1, y1) and (x2, y2) in the distance formula:
Radius = 1/2 * √((3 - 5)^2 + (-1 - 1)^2)
= 1/2 * √((-2)^2 + (-2)^2)
= 1/2 * √(4 + 4)
= 1/2 * √8
= √2
Therefore, the radius of the circle is √2.
Step 3: Writing the equation of the circle
The equation of a circle can be written in the standard form as (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
Plugging in the values we found:
Equation of the circle: (x - 4)^2 + (y - 0)^2 = (√2)^2
(x - 4)^2 + y^2 = 2
So, the equation of the circle is (x - 4)^2 + y^2 = 2.