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).

Answer 1

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).