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.

Question

from Kumon level I math page 193b

Answer 1

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

Since y = x and y = x^2 intersect at points A and B, we can set these two equations equal to each other:

x = x^2

Solve for x:

x^2 - x = 0

x(x - 1) = 0

This gives us x = 0 and x = 1 as the x-coordinates of points A and B.

Substitute these values into y = x to find the y-coordinates:

For point A: A(0, 0)
For point B: B(1, 1)

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

Length OA = sqrt((0 - 0)^2 + (0 - 0)^2) = sqrt(0 + 0) = 0

Length OB = sqrt((1 - 0)^2 + (1 - 0)^2) = sqrt(1 + 1) = sqrt(2)

Therefore, the length of line segment OA is 0 units, and the length of line segment OB is sqrt(2) units.