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I was taking the 2014 AMC 12A test when I came upon #25, which is as follows:

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\leq 1000$?

I got that the directrix of this (rotated) parabola was $y=\frac{3}{4}x-\frac{25}{4}$ but I got stuck there. I was able to understand from the solutions on AoPS and the solution in this video from AoPS that I should rotate the parabola using rectangular to polar coordinate transformations and back. I think that it is a clever way to solve such a problem, but is there any quicker method or formula that when given the focus and directrix (which is tilted), I can easily find the equation of the parabola?

I have also seen and read this previously asked question if anyone was wondering.

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  • $\begingroup$ There are two parabolas that fit the quoted description. $\endgroup$ – amd Jan 7 at 8:20
  • $\begingroup$ Yes for this specific problem but given the directrix there would only be one parabola. $\endgroup$ – Soham Konar Jan 8 at 0:12
  • $\begingroup$ You wrote ”the directrix” when there are two possibilities. Is there more to the problem than what you’ve quoted that would allow you to eliminate one? $\endgroup$ – amd Jan 8 at 0:22
  • $\begingroup$ No there isn't any more information but I think both parabolas could solve the problem because of their symmetry and the absolute value in $|4x+3y|\le 1000$. Ultimately, my question is not about the specific problem I was solving, but actually my desire to generalize the solution to other problems involving tilted parabolas. $\endgroup$ – Soham Konar Jan 8 at 0:49
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One definition of a parabola is that it is the locus of points equidistant from a fixed point (the focus) and a fixed line (the directrix). So, if you know the focus $(x_f,y_f)$ and directrix $ax+by+c=0$, then using the standard formulas for distance to a point and a line, you can write down an equation for the parabola directly: $$(x-x_f)^2+(y-y_f)^2={(ax+by+c)^2\over a^2+b^2}.$$

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