Parametric question (1 Viewer)

[fjitlid]

P(2ap, ap^2) and Q(2ap^2, ap&4) are two points on the parabola x^2=4ay.

I) show that the chord PQ has a gradient m=(1/2)(P^2 + P)

II) Find the minimum gradient of the chord PQ.


i could do part I pretty easily, but am struggling with part II, and the solution makes no sense to me at all. could someone pease help me, and define the minimum gradient in a question like this.


The solution reads...

The minimum occurs when P = (-1/2)/(2x(1/2))
= (-1/2)

then sub -1/2 into the gradient PQ equation to get (-1/8)

 

[pikachu975]

dm/dp = (1/2)(2p+1)
Back to 2 unit, min/max occurs when derivative = 0
2p+1 = 0
p = -1/2

d^2 m/dp^2 = (1/2)(2) = 1 which is always positive so p = -1/2 gives a minimum (concave up)

Sub it back in to get -1/8

 

[fjitlid]

Ahhhh, fair enough.

thanks bro.