When line y= x and parabola y= x squared and y= negative 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

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

Since line y=x intersects the parabola y=x^2, we can set the two equations equal to each other to find the x-coordinates of points A and B:

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

So, x = 0 or x = 1.

When x=0, y = 0 (from y=x), and when x=1, y=1 (from y=x). These are the coordinates of the points O and A respectively.

Similarly, to find point B, we set the equation of the parabola equal to the equation of the line y = -1/4x^2:

x^2 = -1/4x^2
5/4x^2 = 0
x = 0

So, when x=0, y = 0. Therefore, the coordinates of points O and B also coincide.

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

For OA:
OA = sqrt((1-0)^2 + (1-0)^2) = sqrt(2)

For OB:
OB = sqrt((0-0)^2 + (0-0)^2) = 0

Therefore, the length of line segment OA is √2 units and line segment OB is 0 units.