when line y=x+2 and parabola y=x squared intersect at points A and B, find the length of the line segment AB (the part of the line connecting A to B).
To find the points of intersection between the line y=x+2 and the parabola y=x^2, we can set the two equations equal to each other:
x + 2 = x^2
Rearranging to get all terms on one side:
0 = x^2 - x - 2
Factor the quadratic equation:
0 = (x-2)(x+1)
This gives us x = 2 and x = -1 as the points of intersection.
Now we need to find the corresponding y-coordinates for these points:
For x = 2:
y = 2 + 2 = 4
Thus, point A is (2, 4).
For x = -1:
y = -1 + 2 = 1
Thus, point B is (-1, 1).
To find the length of the line segment AB, we can use the distance formula:
AB = sqrt((x2-x1)^2 + (y2-y1)^2)
AB = sqrt((-1-2)^2 + (1-4)^2)
AB = sqrt((-3)^2 + (-3)^2)
AB = sqrt(9 + 9)
AB = sqrt(18)
AB = 3sqrt(2)
Therefore, the length of the line segment AB is 3sqrt(2).