+1557 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.
+129885 Sub the equation of the line into the equation of the circle
(x - 4)^2 + (3x - 2 -5)^2 = 68
(x -4)^2 + ( 3x - 7)^2 = 68
x^2 - 8x + 16 + 9x^2 -42x + 49 = 68
10x^2 - 50x - 3 = 0
Using the Quadratic Formula the solutions for x are
x = (5/2) + sqrt (131/5) / 2
x = (5/2) - sqrt (131/5) / 2
Adding these values and dividing by 2 gives us the x coordinate of the midpoint = 5/2
The y values are 3 (5/2 + sqrt (131/5) / 2) - 2 = 15/2 + 3sqrt (131/5)/2 - 2
And 3(5/2 - sqrt (131/5) /2) -2 = 15/2 - 3sqrt (131/5)/2 - 2
Adding these and dividing by 2 gives us the y coordinate of the midpoint = 11/2
Midpoint = (5/2, 11/2)
