Question

When line y=x-2 and parabola y=-x squared intersect at points A and B, find the length of the line segment AB

Answer 1

To find the points of intersection between the line y=x-2 and the parabola y=-x^2, we need to set these two equations equal to each other:

x - 2 = -x^2

Rearranging the equation, we get:

x^2 + x - 2 = 0

This is a quadratic equation that can be factored as:

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

This gives us two possible values for x: x = -2 and x = 1. Now, we can plug these x-values back into the equation y=x-2 to find the corresponding y-values:

For x = -2:
y = -2 - 2
y = -4

So, the first point of intersection is (-2, -4).

For x = 1:
y = 1 - 2
y = -1

So, the second point of intersection is (1, -1).

Now, we can find the distance between these two points using the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Distance = √((1 - (-2))^2 + (-1 - (-4))^2)
Distance = √((1 + 2)^2 + (-1 + 4)^2)
Distance = √(3^2 + 3^2)
Distance = √(9 + 9)
Distance = √18
Distance ≈ 4.24

Therefore, the length of the line segment AB is approximately 4.24 units.