Question

When line y= x and parabola y= x squared and y=-1/4x squared intersect at points A and B in addition to at origin O as shown below, find the lengths of line segments OA and OB.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the lengths of line segments OA and OB, we need to first find the coordinates of points A and B.

Setting y = x equal to y = x^2, we get:

x = x^2
x^2 - x = 0
x(x-1) = 0
x = 0 or x = 1

Therefore, the points of intersection are (0, 0) and (1, 1).

Next, setting y = -1/4x^2 equal to y = x^2, we get:

-1/4x^2 = x^2
-1 = 4x^2
x^2 = -1/4

This equation has no real solutions, so the parabola and the line do not intersect at any other points besides the ones previously found.

Now, to find the lengths of line segments OA and OB, we can use the distance formula:

For point O(0,0):
OA = √(0^2 + 0^2) = √(0) = 0

For point A(1,1):
OB = √(1^2 + 1^2) = √2

Therefore, the lengths of line segments OA and OB are 0 and √2, respectively.