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.
Answers
GPT 3.5
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.
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.