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