Let there be some parabola, where (0,0), (p,r) and (q,s) are on the curve. (In the image, p=x

_{O}, q=x

_{B}, r=y

_{O}and s=y

_{B}, but it was simpler to use single letters.)

That parabola, like any parabola, is described as y = ax

^{2}+ bx + c, so we can derive

0 = c

r = ap

^{2}+ bp

s = aq

^{2}+ bq

We know q, r and s, and we want to find p.

Putting this into Wolfram Alpha gives us six answers: http://bit.ly/1STUAmi

We can eliminate three of them by stating that q ≠ 0 (because q = 0 means that the "parabola" is actually a vertical line) and another one by stating that a ≠ 0 (because a = 0 means that the "parabola" is actually a straight line).

This leaves us with these two answers:

However, we do not know what b is.

Intuitively, by looking at the image above, I cannot see how there can be more than one solution if p, q, r, s are all real numbers. I can only imagine Wolfram Alpha is (quite correctly) not making the assumption of them being real but I can't find a way of telling it they are real. So I'm a bit stuck - can anyone find b for me, and thus also p?