Parabola Questions (1 Viewer)

[XcarvengerX]

Can you help me with these couple of questions?
1. Show that the locus of the mid-points of chords in the parabola x²=4ay, and which pass through the vertex, is another parabola, x²=2ay.
2. Two points P and Q move on the parabola x²=4ay so that the x coordinates of P and Q differ by a constant value of 2a. What is the locus of M, the mid-point of PQ?

I am confuse with locus concept. I couldn't even draw what the quetions are asking. Thanks in advance.

 

[SoulSearcher]

I'll do the first question

since the chords pass through the vertex of the parabola, one point is (0,0)
let the other point be ( 2at , at² )

so midpoint of the line is ( at , at²/2 )
therefore
x = at and
y = at²/2
x/a = t
therefore
y = a(x/a)² / 2
y = { a * x²/a² } / 2
y = x² / 2a
therefore
x² = 2ay

for the other question, i got up to x = 2y + a(p+q), but i haven't figured out the value of either q or p yet. Ill post as soon as im done

 

[XcarvengerX]

On the first question what I didn't understand before was how can a chord passes through a vertex? But seen your solution, I understand it now. Thanks heaps.