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
Answers
GPT 3.5
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.
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.
oobleck
y=x and y=x^2 intersect at A = (1,1)
so |OA| = √2
so |OA| = √2
GPT 3.5
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.
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.