/Parametrics Problem! (1 Viewer)

[Jackson94]

P and Q are the point t=p and t=q on the parabola x = 2at, y = at^2
a) Find the equations of the normals to the curve P and Q.
b) Prove that p^3 -q^3 = (p-q)(p^2 + pq + q^2)

Thanks in advance guys, obv part a is easy but you need it for part b I presume so I left it in.

 

[Aykay]

Part a:


For part b, is it a hence prove, or just prove? Because it's just a difference of two cubes.

 

[RealiseNothing]

The equations are:

P x+py=2ap+ap^3
Q x+qy=2aq+aq^3 (From the equation of a normal formula)

Minus Q from P:

(x-x) + (py-qy) = (2ap-2aq) + (ap^3 - aq^3)

Simplifying gives:

y(p-q)=2a(p-q)+a(p^3 -q^3)

Rearranging:

p^3-q^3 = (1/a) [y(p-q)-2a(p-q)]

Factoring out p-q from RHS:

p^3-q^3= (1/a) [(y-2a)(p-q)]

Since y=ap^2, substitute this in

p^3-q^3= (1/a) [(ap^2 -2a)(p-q)]

p^3-q^3= (1/a) [a(p^2 -2)(p-q)]

Now you can cancel out the a's:

p^3-q^3= (p-q)(p^2 -2)

Not sure what to do from here though,all we need to do is turn that -2 into pq+q^2 somehow.

 

[SpiralFlex]

(Intersection x coordinate of the normal.)



Substitute that into your equation.