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.
from Kumon level I math page 193b
In the given scenario, the points of intersection of the lines y = x and the parabolas y = x^2 and y = -1/4x^2 are the points A and B, respectively. In addition, they intersect at the origin O (0,0).
To find the lengths of line segments OA and OB, we first need to determine the coordinates of points A and B.
For point A, we set y=x and y=x^2 equal to each other:
x = x^2
x^2 - x = 0
x(x - 1) = 0
x = 0 or x = 1
So point A has coordinates (0, 0) and (1, 1).
For point B, we set y = x and y = -1/4x^2 equal to each other:
x = -1/4x^2
4x^2 = -x
4x^2 + x = 0
x(4x + 1) = 0
x = 0 or x = -1/4
So point B has coordinates (0, 0) and (-1/4, -1/4).
Now, we can calculate the lengths of line segments OA and OB:
OA = √((x - x1)^2 + (y - y1)^2)
For point A (0, 0):
OA = √((0 - 0)^2 + (0 - 0)^2) = √(0 + 0) = √0 = 0
For point A (1, 1):
OA = √((1 - 0)^2 + (1 - 0)^2) = √(1 + 1) = √2
Similarly, for point B (-1/4, -1/4):
OB = √((-1/4 - 0)^2 + (-1/4 - 0)^2) = √(1/16 + 1/16) = √(1/8) = √(2/16) = √(1/8) = √1 * √(1/8) = 1 * 1/√8 = 1/√8 = √8/8 = √2/4 = 1/2
Therefore, the lengths of line segments OA and OB are √2 and 1/2, respectively.