A circle with center C(4,6) has a diameter that terminates at A(-1,y) and B(x,8). Find the values of x and y.
C is the center, so
4 = (-1+x)/2
6 = (y+8)/2
...
To find the values of x and y, we can use the properties of a circle.
1. The midpoint of a diameter of a circle is the center of the circle.
Therefore, the coordinates of the midpoint M of segment AB are:
M = [(x + (-1))/2, (8 + y)/2] = [(x - 1)/2, (y + 8)/2]
2. Since A and B are the endpoints of the diameter, the midpoint M must coincide with the center C of the circle:
(x - 1)/2 = 4 and (y + 8)/2 = 6
Solving these equations will give us the values of x and y:
(x - 1)/2 = 4 -> x - 1 = 8 -> x = 9
(y + 8)/2 = 6 -> y + 8 = 12 -> y = 4
Therefore, the values of x and y are 9 and 4, respectively.