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.
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:
So mirroring $F$ across $a$ results in a point $F'$, and the line $DF'$ is the directrix.
$\qquad$
Another useful parabola feature is:
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.