Locus Problem (1 Viewer)

[bell531]

Ok, I can't seem to get this out, any help appreciated:

The chord PQ is a focal chord of the parabloa X^2 = 4ay where P=(2ap, ap^2) and Q=(2aq, aq^2). Find the eqn of the locus of the midpoint of PQ.

 

[Drongoski]

Ok, I can't seem to get this out, any help appreciated:

The chord PQ is a focal chord of the parabola X^2 = 4ay where P=(2ap, ap^2) and Q=(2aq, aq^2). Find the eqn of the locus of the midpoint of PQ.

Eqn of chord PQ: y = (1/2)(p+q)x - apq

For focal chord, chord passes thru focus = F(0,a)

Therefore: a = (1/2)(p+q) x 0 - apq ==> pq = -1

Mid-pt given by: x = (2ap+2aq)/2 = a(p+q)

y = (ap^2 + aq^2)/2 = a[(p+q)^2 - 2pq]/2

= a[(p+q)^2 + 2]/2 = a[(x/a)^2 + 2]/2

Therefore: y = (x^2 + 2a^2)/(2a)

Therefore x^2 = 2ay - 2a^2 = 4(a/2) (y - a)


That means the locus is: x² = 2a(y - a)

i.e. another parabola focal length 0.5a, focus (0, 1.5a), vertex (o,a) and directrix y = (1/2) a


Edit: I typed out the above soln but lost it a few hours ago!

 

[bell531]

I didn't get it from that book, and I don't have it, but what I'm trying to figure out is what to do (to get the eqn) once I've values for x and y in terms of a and q. Solutions would be helpful, if only the later few steps.


EDIT: DW, I got it finally. Was a decent question too. Thanks for your help.

 

[Drongoski]

I didn't get it from that book, and I don't have it, but what I'm trying to figure out is what to do (to get the eqn) once I've values for x and y in terms of a and q. Solutions would be helpful, if only the later few steps.


EDIT: DW, I got it finally. Was a decent question too. Thanks for your help.

See my edited version. I typed out the whole solution but lost it. So I've retyped it. Hope it's clearer now.