+1248 A line and a circle intersect at $A$ and $B,$ as shown below. Find the coordinates of the midpoint of $\overline{AB}$.
The circle is x^2 + y^2 = 6. The line is x + y = 2.
+129771 Equation of line
y = 2 -x
Sub this into equation of circle
x^2 + ( 2 - x)^2 = 6
x^2 + x^2 - 4x + 4 = 6
2x^2 - 4x = 2
x^2 -2x = 1
x^2 -2x + 1 = 1 + 1
(x -1)^2 = 2
x - 1 = sqrt (2) and x -1 = -sqrt (2)
x = 1 + sqrt (2) and x = 1 -sqrt (2)
y = 1 -sqrt (2) and y = 1 +sqrt (2)
A = (1 + sqrt (2) , 1 -sqrt (2) ) B = (1-sqrt (2) , 1 + sqrt (2) )
Midpoint = (1,1)
