Parabola Question (1 Viewer)

[Glen88]

Two points P and Q move along the parabola y=x³/4a in such a way that the x-coordinates of P and Q differ by the constant amount 2a. Find the equation of the locus of the midpoint of the chord PQ, and describe in words this locus.

 

[Trev]

You must mean y=x²/4a
P(2ap,ap²), Q(2aq,aq²)
Let the x-coordinate of Q be the difference of the x-coordinate of P so Q(2ap-2a,aq²).
Midpoint:
[a(2p-1)/2 , a(p²+q²)/2]
Since 2aq=2ap-2a then q=p-1.
Y-coordinate of the midpoint;
y=a(p²+q²)/2
2y/a=p²+(p-1)²
2y/a=2p²-2p+1
y/a=p²-p+1/2
y/a=(p-1/2)²+3/4
y/a-3/4=(p-1/2)² -- (1)
X-coordinate of midpoint;
x=a(2p-1)/2
x/a=p-1/2 -- (2)
(2) into (1) gives
y/a-3/4=(x/a)² which is the locus, now describe it.

 

[Glen88]

y/a=p²-p+1/2 <-- How did you go from here
y/a=(p-1/2)²+3/4 <-- To Here.

 

[EmmR]

y/a=p²-p+1/2 <-- How did you go from here
y/a=(p-1/2)²+3/4 <-- To Here.

Its called completing the square.

=p²-p+1/2
=p²-p+1/4+1/2-1/4
you add a quarter and then take away
because you gotta choose a number that fits the rule
(x+b)²= x²+2b+b²

and b² happens to be 1/4

soo from ..

=p²-p+1/4+1/2-1/4
it becomes

=(p-1/2)²+1/2-1/4
=(p-1/2)²+3/4


^^ this is the first time i replyed to any thing.. hoped it helped

 

[Rax]

He completed the square

LOL Omg EmmR's reply appeared as I did mine, meh my bad

And yep she explains it nicely

Carry on