The line y = (x - 2)/2 intersects the circle x^2 + y^2 = 8 at A and B. Find the midpoint of AB. Express your answer in the form (x,y).
+23246 First, let's find where the line y = (x - 2)/2 intersects the circle x2 + y2 = 8 by substituting the first value for y into the second equation:
---> x2 + [ (x-2)/2 ]2 = 8
Expand the second term:
---> x2 + ( x2 - 4x + 4 ) / 4 = 8
Multiply all term by 4:
---> 4x2 + x2 - 4x + 4 = 32
Simplify the left side and subtract 32 from both sides:
---> 5x2 - 4x - 28 = 0
Factor (or use the quadratic formula):
---> (5x - 14)(x + 2) = 0
Solve for x and put this value into the linear equation to find the corresponding y-value:
---> 5x - 14 = 0 ---> x = 14/5 = 2.8 and y = 0.4
or x + 2 = 0 ---> x = -2 and y = -2
The points of intersection are: (2.8, 0.4) and (-2, -2)
Use the midpoint formula to find the midpoint of these two points of intersection:
Their midpoint is: (0.2, -0.8)
