Question
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.
Answers
GPT 3.5
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.
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.