When line y=2x+2 and parabola y=-x squared - 4x-3 intersect at points A and B, find the length of the line segment AB
To find the points of intersection A and B, we need to solve the system of equations formed by the line y=2x+2 and the parabola y=-x^2-4x-3.
Setting the two equations equal to each other:
- x^2 - 4x - 3 = 2x + 2
- x^2 - 6x - 5 = 0
- (x - 1)(x + 5) = 0
So, x = 1 or x = -5.
When x = 1:
- y = 2(1) + 2
- y = 4
So, one point of intersection is A(1,4).
When x = -5:
- y = 2(-5) + 2
- y = -8
So, another point of intersection is B(-5, -8).
To find the length of the line segment AB, we use the distance formula:
d = sqrt((x2-x1)^2 + (y2-y1)^2)
d = sqrt((-5-1)^2 + (-8-4)^2)
d = sqrt((-6)^2 + (-12)^2)
d = sqrt(36 + 144)
d = sqrt(180)
d = 6 sqrt(5)
Therefore, the length of the line segment AB is 6√5 units.