When line y= 2x and parabola y= x squared and y= 2x squared are as shown below, solve the following problems

Question

from Kumon level I math page 191b
Find the point of intersection A, which is where line y=2x and parabola y=x squared intersect in addition to at origin O.

Find the point of intersection B, which is where line y=2x and parabola y=2x squared intersect in addition to at origin O.

Find the length of line segments OA and OB.

Answer 1

To find the point of intersection A, we first need to set the equations y=2x and y=x^2 equal to each other:

2x = x^2
x^2 - 2x = 0
x(x-2) = 0

Therefore, x = 0 or x = 2. Substituting x back into y=2x, we get y=0 when x=0 and y=4 when x=2. So the point of intersection A is (2,4).

To find the point of intersection B, we set the equations y=2x and y=2x^2 equal to each other:

2x = 2x^2
2x^2 - 2x = 0
2x(x-1) = 0

Therefore, x=0 or x=1. Substituting x back into y=2x, we get y=0 when x=0 and y=2 when x=1. So the point of intersection B is (1,2).

To find the length of line segment OA, we can use the distance formula:

OA = sqrt((2-0)^2 + (4-0)^2) = sqrt(4 + 16) = sqrt(20) = 2sqrt(5).

To find the length of line segment OB, we can use the distance formula:

OB = sqrt((1-0)^2 + (2-0)^2) = sqrt(1 + 4) = sqrt(5).

Therefore, the length of line segment OA is 2sqrt(5) and the length of line segment OB is sqrt(5).