DETERMINE the equation of the circle that has the following three points on its circumference: A(1,5), B(3,7) and C(5,5).
(x-k)^2+ (y-h)^2 = r^2
so
(1-k)^2 + (5-h)^2 = r^2
(3-k)^2 + (7-h)^2 = r^2
(5-k)^2 + (5-h)^2 = r^2
Now before I go off and solve those for k,h and r, sketch a graph
Notice that two points, A and C are at the same height,5
The third point,B, is half way between them (3 is halfway between 1 and 5)
I conclude that the center of the circle is on x = 3
so
k = 3
Onward
(1-3)^2 + (5-h)^2 = r^2
(3-3)^2 + (7-h)^2 = r^2
(5-3)^2 + (5-h)^2 = r^2
that second equation is pretty easy now.
(7-h)^2 = r^2
the first and third are actually the same now that we know what k isso use the first and the second
4 + (5-h)^2 = r^2
(7-h)^2 = r^2
29 - 10 h + h^2 = r^2
49 -14 h + h^2 = r^2
----------------------
-20 +4 h = 0
h = 5
well I guess you can take it from there
Thank you!
To determine the equation of a circle that passes through three points, we can use the formula for the circle, which is:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the center coordinates of the circle, and r represents the radius.
Here's how we can determine the equation:
1. Find the midpoint of any two points. Let's find the midpoint of points A(1, 5) and B(3, 7).
- Midpoint formula: [(x1 + x2) / 2, (y1 + y2) / 2]
- Midpoint of AB: [(1 + 3) / 2, (5 + 7) / 2] = (2, 6)
2. Find the midpoint of the remaining two points. Let's find the midpoint of points A(1, 5) and C(5, 5).
- Midpoint of AC: [(1 + 5) / 2, (5 + 5) / 2] = (3, 5)
3. Find the equation of the perpendicular bisector of AB and AC. This line will pass through the center of the circle.
- Slope of AB: (7 - 5) / (3 - 1) = 2/2 = 1
- Slope of AC: (5 - 5) / (5 - 1) = 0/4 = 0
- Since the slope of a perpendicular line is the negative reciprocal of the original slope, the slope of the perpendicular bisector of AB is -1.
- Using the midpoint (2, 6) and the slope -1, we can determine the equation of the perpendicular bisector of AB:
- y - 6 = -1(x - 2)
- y - 6 = -x + 2
- y = -x + 8
- Using the midpoint (3, 5) and the slope -1, we can determine the equation of the perpendicular bisector of AC:
- y - 5 = -1(x - 3)
- y - 5 = -x + 3
- y = -x + 8
4. Find the intersection point of the two perpendicular bisectors. This point will be the center of the circle.
- Solving the two equations, we get:
- -x + 8 = -x + 8
- y = 8
- The intersection point of the perpendicular bisectors is (x, y) = (any x coordinate, 8).
Let's take x = 0, which gives us (0, 8) as the center of the circle.
5. Calculate the radius of the circle. This can be done by finding the distance between the center and any of the given points. Let's calculate the distance between (0, 8) and point A(1, 5).
- Distance formula: sqrt((x2 - x1)^2 + (y2 - y1)^2)
- Distance between (0, 8) and A(1, 5): sqrt((1 - 0)^2 + (5 - 8)^2) = sqrt(1 + 9) = sqrt(10)
6. Finally, substitute the values of the center and radius into the circle equation (x - h)^2 + (y - k)^2 = r^2:
- (x - 0)^2 + (y - 8)^2 = (sqrt(10))^2
- x^2 + (y - 8)^2 = 10
Therefore, the equation of the circle that passes through points A(1, 5), B(3, 7), and C(5, 5) is x^2 + (y - 8)^2 = 10.