When line y=-3x+7 and parabola y=x squared -4x+1 intersect at points A and B, find the length of the line segment AB

Answer 1

To find the points of intersection (A and B), we need to set the two equations equal to each other:

-3x + 7 = x^2 - 4x + 1

Rearranging the terms, we get a quadratic equation:

x^2 - x - 6 = 0

Factorizing, we get:

(x - 3)(x + 2) = 0

Therefore, the points of intersection are x = 3 and x = -2.

To find the y-coordinates for these points, we substitute these values back into either of the equations. Using y = -3x + 7:

For x = 3, y = -3(3) + 7 = -2
For x = -2, y = -3(-2) + 7 = 13

Therefore, the points of intersection are A(3, -2) and B(-2, 13).

Using the distance formula, we can find the length of the line segment AB:

AB = sqrt((3 - (-2))^2 + (-2 - 13)^2)
AB = sqrt(5^2 + (-15)^2)
AB = sqrt(25 + 225)
AB = sqrt(250)
AB = 5√10

So, the length of the line segment AB is 5√10.