Question

When line y= x and parabolas 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 length of line segment OA and OB with the correct graph

Answer 1

y=x and y=x^2 intersect at A = (1,1)
so |OA| = √2

Answer 2

To find the length of line segment OA and OB, we first need to find the coordinates of points A and B.

Given that the line y = x intersects with the parabolas y = x^2 and y = -1/4x^2, we can set up the following equations to find the coordinates of points A and B:

1. For point A:
x = x (from the line equation y = x)
y = x^2 (from the parabola y = x^2)

So, x = x and y = x^2. Substituting y = x^2 into the line equation, we get:
x^2 = x
x^2 - x = 0
x(x - 1) = 0
x = 0 or x = 1

For x = 0, y = 0^2 = 0. So, point A is at (0, 0).

2. For point B:
x = x (from the line equation y = x)
y = -1/4x^2 (from the parabola y = -1/4x^2)

So, x = x and y = -1/4x^2. Substituting y = -1/4x^2 into the line equation, we get:
-1/4x^2 = x
-1/4x^2 - x = 0
x(-1/4x - 1) = 0
x = 0 or x = -4

For x = 0, y = 0^2 = 0. So, point B is at (0, 0).
For x = -4, y = -1/4(-4)^2 = -4. So, point B is also at (-4, -4).

Now, we can calculate the length of line segment OA and OB:

1. Length of OA:
OA = √((x-0)^2 + (y-0)^2)
OA = √((0-0)^2 + (0-0)^2) = √(0 + 0) = √0 = 0

2. Length of OB:
OB = √((-4-0)^2 + (-4-0)^2)
OB = √((-4)^2 + (-4)^2) = √(16 + 16) = √32 ≈ 5.66

Therefore, the length of line segment OA is 0 units and the length of line segment OB is approximately 5.66 units.

Answer 3

Apologies for the confusion in the previous response. Let's correct the calculations using the correct point of intersection between the line y = x and the parabola y = x^2.

Given that y = x and y = x^2 intersect at point A, which is (1,1), the length of line segment OA can be calculated as follows:

OA = √((x2 - x1)^2 + (y2 - y1)^2)
Where:
x1 = 0, y1 = 0 (origin O)
x2 = 1, y2 = 1 (point A)

OA = √((1 - 0)^2 + (1 - 0)^2)
OA = √(1^2 + 1^2)
OA = √(1 + 1)
OA = √2

Therefore, the length of line segment OA is √2 units, which is approximately 1.41 units. Thank you for pointing out the error, and I appreciate your understanding.