p(x),f(x),g(x) are parabolas with positive coefficient, each pair of parabolas got common rational root, how to proof that p(x)+f(x)+g(x) have a rational root?

a≈13.68cm

The vertex is always halfway between the focus and the directrix, and the parabola always curves away from the directrix, so I'll do a quick graph showing the focus, the directrix, and a rough idea of where the parabola.

So the vertex, exactly between the focus and directrix, must be at (h, k) = (1, 
–2). The absolute value of p is the distance between the vertex and the focus and the distance between the vertex and the directrix. (The sign on ptells me which way the parabola faces.) Since the focus and directrix are two units apart, then this distance has to be one unit, so | p | = 1.Since the focus is to the left of the vertex and directrix, then the parabola faces left (as I'd shown in my picture) and I get a negative value for p: p = –1. Since this is a "sideway" parabola, then the y part gets squared, rather than the x part. So the conics form of the equation must be:(y – (–2))2 = 4(–1)(x – 1), or (y + 2)2 = –4(x – 1)

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