Parabolas and directrices

Parabolas are on the borderline between ellipses and hyperbolas. They can be approximated by ellipses with eccentricities just below 1, or by hyperbolas with eccentricity just above 1. Either way, the foci are getting farther and farther apart as the eccentricity approaches 1, and the limiting shape cannot be described by two foci.

Instead, a parabola is determined by a single focus and a line called a directrix.

Definition of a parabola:Let $L$ be the distance to the focus and let $D$ be the distance to the directrix. Then a parabola is the set of all points with $L=D.$

Example 1: Find the equation of a parabola with a focus at $(0,1)$ and a directrix on the $x$-axis. Solution: We have $D=|y|$ and $L = \sqrt{x^2+(y-1)^2}$. Our equations $L=D$ then become:

\begin{eqnarray*}\sqrt{x^2+(y-1)^2}&=&|y|\cr x^2 + (y-1)^2 & = & y^2 \cr x^2 + y^2 -2y + 1 &=& y^2 \cr x^2 + 1 &=& 2y \cr y & = & \frac{x^2+1}{2}.\end{eqnarray*}

Example 2: Find the equation of a parabola with a focus at $(0,p)$ and a directrix at $y=-p$. Solution: Now we have $D=|y+p|$ and $L = \sqrt{x^2+(y-p)^2}$. Our equations $L=D$ are now:

\begin{eqnarray*}\sqrt{x^2 + (y-p)^2} & = & y+p \cr x^2 + (y-p)^2 &=& (y+p)^2 \cr x^2 + y^2 -2py + p^2 &=& y^2 + 2py + p^2 \cr x^2 &=& 4py \cr y & = & \frac{x^2}{4p}.\end{eqnarray*} Similarly, the horizontal parabola $x = \frac{y^2}{4p}$ has focus at $(p,0)$ and directrix at $x=-p$.