+1323 A line and a circle intersect at A and B. Find the coordinates of the midpoint of \overline{AB}.
The ilne is y = 3x - 2, and the circle is (x - 4)^4 + (y - 5)^2 = 68.
+129829 y=3x + 2 (1)
(x-4)^2 + (y -5)^2 = 68
The center of the circle is (4, 5)
The slope of the given line is 3
The slope of a perpendicular line is -1/3
Using (4,5) and writing an equation for this perpendicular line gives us
y = -(1/3)(x -4) + 5
y = (-1/3)x + 4/3 + 5
y = (-1/3)x + 19/3
Finding the x intersection of these two lines
3x + 2 = (-1/3)x + 19/3
9x + 6 = -x + 19
10x = 13
x = 13/10
y = 3(13/10) + 2 = 47/4
The coordinates of the midpoint = (13/10, 59/10)
