Parametric Locus Problem (1 Viewer)

[ordnaiv]

PQ is a chord of the parabola x^2 = 4ay such that the abscissa of P is twice that of Q. Show that the locus of the midpoint of PQ is the parabola 5x^2 = 18ay.

 

[Riviet]

P(2ap,ap²) and Q(2aq,aq²).

Abscissa means x-coordinate.

So 2ap=2(2aq)
p=2q (1)
p²=4q² (2)
Now midpoint of PQ is

{(2ap+2aq)/2 , (ap²+aq²)/2}
={a(p+q),a/2.(p²+q²)}

Now p+q=x/a by rearranging the x-coordinate of the midpoint.

.'.2q+q=x/a, using (1)
q=x/3a
q²=x²/9a² (3)

Now y=a/2.(p²+q²)
y=a/2.(4q²+q²), using (2)
y=a/2.(5q²)
y=(a/2).(5x²/9a²), using (3)

Rearranging this yields 5x²=18ay.