A parabola puzzle

[Keiji]

This problem came up during some game development, surprisingly enough.

FreshPaint-7-2016.01.16-11.39.02.png

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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 = ax² + bx + c, so we can derive

0 = c
r = ap² + bp + c
s = aq² + bq + c

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:

answers.png (10.03 KiB) Viewed 1690 times

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?

[Keiji]

I just realised my mistake - (p,r) is at the top of the curve, but I never wrote that down formally. Whoops