A challenge (kinda)

[Mao]

To current year 11/12 kids, this problem was quite interesting:

There are three arbitrary points P, Q and R in space. Points P and Q are on the x axis equidistant from the origin. The x coordinate of point R lies between P and Q, the y coordinate of R is unspecified. It is always possible to construct an ellipse through these three points.

Let be the displacement vector from P to Q, and similarly define other displacement vectors. We can show that the ellipse will always be in the form of

, where C is some constant depending on the position of R.

Find an expression for C in terms of the displacement vectors (do not use scalars, i.e. do not say "let , ....").

[theuncle]

Ive done the working in my book but i thought i'd try the answer first incase it's wrong! (r is the position vector of R)
does

[kyzoo]

I think your answer is correct only if R is on the y-axis.

[theuncle]

Ok, so i'll explain how i got it...
we know that
1-
also, 2-
from here it is a matter of solving simultaneous equations by subbing in known points of x and y
we know that when

subbing this into equation 2,
therefore...

this also holds for y=0
as far as I can see... I've given an equation of an ellipse with (0,0) centre that gives correct values when x=0 and y=0,
I figured no other equation could give an ellipse that had the same max x and y values... but it wouldn't be the first time i'm wrong

[kyzoo]

Your first equation describes a circle, but R, P, and Q form an ellipse

[theuncle]

It would be a circle if r was a constant, but because r changes, ie. the radius of the circle changes... the equation i gave was infact that of an ellipse.

...C is constant, but when we fix r at a point,
holds for that point. And that's all that matters

[Mao]

I have no idea if my expression will simplifies to yours when R lies on the y axis. But regardless, that assumption cannot be made. The point R is any point lying between the x coordinates of P and Q with an arbitrary y coordinate.

[googoo]

Let R be vector OR, r be vector OQ and u be unit vector parallel to vector OQ.

C = (|R|^2 - (R.u)^2) / (|r|^2 - (R.u)^2)

[Mao]

Displacement vectors? You've given the answer in terms of position vectors.

[googoo]

R = vectorPR - r, r = 0.5vectorPQ

By the way, R and r are not position vectors. I considered them as displacement vectors from O to R and Q respectively.