Find the equation of the circle whose diameter has the end point A(5,4) and B(1,4).
The centre must be the midpoint of AB, which is
( (5+1)/2 , (4+4)/2) or (3,4)
so the equation must be (x-3)^2 + (y-4)^2 = r^2
plug in one of the given points to find r^2 and you are done.
To find the equation of the circle, we can first calculate the center of the circle using the midpoint formula, and then determine the radius.
Step 1: Calculating the center of the circle.
The midpoint formula states that the coordinates of the midpoint between two points (x1, y1) and (x2, y2) are given by:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
In this case, the two endpoints of the diameter are A(5, 4) and B(1, 4). Applying the midpoint formula, we get:
Midpoint = ((5 + 1) / 2, (4 + 4) / 2)
Midpoint = (6 / 2, 8 / 2)
Midpoint = (3, 4)
So, the center of the circle is C(3, 4).
Step 2: Determining the radius of the circle.
The distance formula can be used to find the distance between the center of the circle and one of the endpoints of the diameter.
The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, we can calculate the distance between C(3, 4) and A(5, 4) using the distance formula:
Distance = sqrt((5 - 3)^2 + (4 - 4)^2)
Distance = sqrt(2^2 + 0^2)
Distance = sqrt(4 + 0)
Distance = sqrt(4)
Distance = 2
Thus, the radius of the circle is 2.
Step 3: Writing 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
In this case, the center of the circle is C(3, 4), and the radius is 2. Substituting these values into the equation, we get:
(x - 3)^2 + (y - 4)^2 = 2^2
(x - 3)^2 + (y - 4)^2 = 4
Therefore, the equation of the circle is (x - 3)^2 + (y - 4)^2 = 4.