+1858 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.
+129653 (x - 4)^2 + (y -5)^2 = 68
This is a circle with a center at (4,5)
A perpepdicular line to the given one passing through the center amd intersecting the given line will give us the midpoint of chord AB
Equation of perpendicular line
y =(-1/3)(x - 4) + 5 set this equal to the given line
3x -2 = (-1/3)(x - 4) + 5 mulriply through by 3
9x -6 = - (x - 4) + 15
10x = 25
x = 25/10 = 2.5
y = 3(2.5) -2 = 5.5
Midpoint = (2.5 , 5.5)
