Locus Problem (1 Viewer)

[GaDaMIt]

Two points P(2ap, ap^2) and Q(2aq, aq^2) where p > q move along the parabola x^2 = 4ay. At all times the x coordinates of P and Q differ by 2a

a) find the midpoint M of chord PQ, and the Cartesian equation of its locus
b ) Give a geometrical description of this locus

ok .. ive done M {a(p + q), 1/2a (p^2 + q^2)}

for the 2nd part of part a)..

im getting y = 1/2x(p + q) - apq...

answer is x^2=4a(y-a/4)

Im guessing this has something to do with the conditions specified in the question... "p > q" and "At all times the x coordinates of P and Q differ by 2a"

please explain this to me.. thanks

 

[Sober]

We know that the x-coordinates differ by 2a so:
2ap-2aq = 2a
(1) p-q = 1

You are correct with M{a(p + q), 1/2a (p² + q²)} so:

x = a(p+q)
(2) (p+q) = x/a

y = a(p²+q²)/2
= a((p-q)² + 2pq)/2
= a(1+2pq)/2 from (1)
(3) 2pq = (2y/a) - 1

y = a(p²+q²)/2
= a((p+q)² - 2pq)/2
= a((x/a)² - (2y/a) + 1)/2 from (2) and (3)

Simplifying...

2y = x²/a - 2y + a
x² = 4ay - a² = 4a(y-a/4)