[Math] Find Vertex when Focus and point on directrix of Parabola is given.

algebra-precalculusanalytic geometryconic sections

Focus is $(2,3)$, point on directrix is $(-3,2)$. Parabola touches $x$-axis. Find vertex.

I would be thankful if someone could help me with this problem.

Best Answer

Here is a geometric way; I leave it to you to translate it to algebra.

You are given the focus $F$, a point $D$ on the directrix and a tangent $a$. For a parabola we have:

  1. Mirroring the focus across a tangent yields a point on the directrix.

So mirroring $F$ across $a$ results in a point $F'$, and the line $DF'$ is the directrix.

$\qquad$Geometric solution

Another useful parabola feature is:

  1. The parabola vertex lies halfway between the focus and the closest point of the directrix.

Therefore consider a straight line through $F$ that intersects the directrix orthogonally in a point $P$. The midpoint between $P$ and $F$ is the sought vertex $V$.

With the given coordinates, $P$ happens to land on the tangent $a$, but that is mere coincidence.

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