When line y=x+7 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, we need to solve the two equations simultaneously:
y = x + 7 and y = x^2 + 4x + 3
Setting the two equations equal to each other, we get:
x + 7 = x^2 + 4x + 3
Rearranging the equation, we get:
x^2 + 4x + 3 - x - 7 = 0
x^2 + 3x - 4 = 0
(x - 1)(x + 4) = 0
Therefore, x = 1 or x = -4
Substitute these values of x back into the equation y = x + 7 to find the corresponding y values:
When x = 1, y = 1 + 7 = 8
When x = -4, y = -4 + 7 = 3
So the points of intersection are (1, 8) and (-4, 3).
To find the length of the line segment AB between these two points, we use the distance formula:
Length AB = sqrt[(x2 - x1)^2 + (y2 - y1)^2]
Length AB = sqrt[(1 - (-4))^2 + (8 - 3)^2]
Length AB = sqrt[(5)^2 + (5)^2]
Length AB = sqrt[25 + 25]
Length AB = sqrt(50)
Length AB = 5*sqrt(2)
Therefore, the length of the line segment AB is 5*sqrt(2) units.