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.
To find the lengths of line segments OA and OB, we first need to find the coordinates of points A and B where the line y=x intersects the parabolas y=x^2 and y=-1/4x^2.
We start by setting y=x equal to y=x^2:
x = x^2
x^2 - x = 0
x(x - 1) = 0
So, x = 0 or x = 1.
Thus, the point of intersection for y=x and y=x^2 are (0,0) and (1,1).
Next, we set y=x equal to y=-1/4x^2:
x = -1/4x^2
Multiplying both sides by 4x:
4x^2 = -x
4x^2 + x = 0
x(4x + 1) = 0
So, x = 0 or x = -1/4.
Thus, the point of intersection for y=x and y=-1/4x^2 are (0,0) and (-1/4, -1/4).
Now, we can calculate the lengths of line segments OA and OB:
OA = √((1 - 0)^2 + (1 - 0)^2) = √(1 + 1) = √2
OB = √((-1/4 - 0)^2 + (-1/4 - 0)^2) = √((1/16) + (1/16)) = √(1/8) = √(2/8) = √(1/4) = 0.5
Therefore, the length of line segment OA is √2 and the length of line segment OB is 0.5.