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 length of line segment OB.

Answer 1

To find the length of line segment OB, we first need to find the coordinates of point B where the parabola y = x^2 and the line y = -1/4x^2 intersect.

Given that the line y = x intersects the parabola y = x^2, we can set the equations equal to each other:

x = x^2

Rearranging gives us:

x^2 - x = 0

Factoring out an x gives:

x(x - 1) = 0

So, the x-coordinates of the intersection points are x = 0 and x = 1.

At point B, the line is represented by the equation y = 1/4x and the parabola is represented by y = -1/4x^2.

Plugging x = 1 into the equations, we find the y-coordinate of point B on the line:

y = 1/4(1) = 1/4

And on the parabola:

y = -1/4(1)^2 = -1/4

So, point B has coordinates (1, 1/4).

The distance between points O and B can be calculated using the distance formula:

Distance OB = √((x2 - x1)^2 + (y2 - y1)^2)
Distance OB = √((1 - 0)^2 + (1/4 - 0)^2)
Distance OB = √(1 + 1/16)
Distance OB = √(17/16)
Distance OB = √17/4
Distance OB = √17/2

Therefore, the length of line segment OB is √17/2.