Hmna wrote:In the xy-plane, the parabola with equation y = -(x + 3)² intersects the line with equation y = 4x at two points, P and Q. What is the length of PQ̅̅̅̅?

A) 40
B) 8√17
C) 50
D) 10√5

If two lines intersect, then at the point of intersection, the two lines share the same solution to the given equations.

Given: y = 4x and y = -(x + 3)²
Take y = -(x + 3)² and replace y with 4x to get: 4x = -(x + 3)²
Expand right side: 4x = -(x² + 6x + 9)
Multiply both sides by -1 to get: -4x = x² + 6x + 9
Rearrange to get: x² + 10x + 9 = 0
Factor: (x + 1)(x + 9) = 0
So, x = -1 or x = -9

If x = -1, then plug x = -1 into the equation y = 4x to get y = -4
So, (-1, -4) is one point of intersection

If x = -9, then plug x = -9 into the equation y = 4x to get y = -36
So, (-9, -36) is one point of intersection

What is the length of PQÌ…Ì…Ì…Ì…?
In other words, what is the distance between the two points of intersection, (-9, -36) and (-1, -4)
Use the distance formula to get: distance = √[(-1 - -9)² + (-4 - -36)²]
= √[8² + 32²]
= √[1088]
= 8√17

Answer: B

Cheers,
Brent