[Math] Equation of a parabola given by 2 points and a focus

analytic geometryconic sections

I want to solve this problem. I know a focus of a parabola: $F=(7,-3)$ and 2 points which parabola lies on $A=(5,7/3)$ and $B=(5,-25/3)$. I also know y-coordinate of vertex of a parabola: $V=(?,-3)$. I have to find vertex equation of a parabola. Thanks for your help.

Best Answer

Two points on a parabola and its focus just fix a pair of parabolas, and we cannot simply assume that their axis will be parallel/perpendicular to the $y$-axis. Let $A,B$ be our points, $F$ our focus. Let $A',B'$ be the projections of $A,B$ on the directrix: clearly, $A'$ belongs to the circle $\Gamma_A$ centered at $A$ through $F$, $B'$ belongs to the circle $\Gamma_B$ centered at $B$ through $F$. The directrix is a common tangent to $\Gamma_A$, $\Gamma_B$, so it goes through the exterior homothetic centre $H$ of $\Gamma_A,\Gamma_B$. $H$ obviously lies on $AB$, and its position on the $AB$ line just depends on the radii of $\Gamma_A$, $\Gamma_B$, i.e. $AF$ and $BF$.

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Summarizing:

  1. Find $AF,BF$ through the Pythagorean theorem;
  2. Use the previous informations to locate $H\in AB$;
  3. One of the tangents to $\Gamma_A$ through $A$ is the directrix of the parabola;
  4. Given the focus and the directrix, there is nothing else to locate: the vertex is just the midpoint of the segment joining $F$ with its projection on the directrix.
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