Definition of a Parabola
Using the Focus and Directrix

Debra Woods
November, 1994

Initialization

Off[General::Spell];
Off[General::Spell1];

Purpose

Previously we have seen that the function f(x) = a (x^2) + b x + c generates a parabola.  The purpose of this lesson is to generate a parabola by using the definition of a parabola.

Definition of a parabola

A parabola is the set of all points P in the plane that are equally distant from a fixed point F (focus) and a fixed line L (directrix) not containing F.

Let's begin by examining a parabola of the form x\(TraditionalForm\`\^2\) = 4 p y.  The focus of the parabola is (0, p) and the directrix is the line y = -p.

The focus of a parabola using the f(x) = ax\(TraditionalForm\`\^2\) formula is the 
point (0, 1/4 a).

The directrix of this parabola in the form f(x) = a x\(TraditionalForm\`\^2\) is the line 
y =  -1/4a.

Run the following cell to see a picture of a parabola along with it's focus and directrix.

Clear[f,x,a,p1,focus];
f[x_] = 1/12 x^2;
p1=Plot[{-3,f[x]},{x,-7,7},
	  PlotStyle->{{RGBColor[0,1,0],Thickness[0.015]},
	             {RGBColor[0,0,1],Thickness[0.01]}},
	    	     AspectRatio->1,
	    	     PlotRange->{{-6,6},{-6,6}},
	     DisplayFunction->Identity];
focus=Graphics[{PointSize[0.025],RGBColor[0,1,0],Point[{0,3}]}];
Show[p1,focus,DisplayFunction->$DisplayFunction];

The above parabola is actually the parabola x\(TraditionalForm\`\^2\) = 12 y, or 
y = 1/12 x\(TraditionalForm\`\^2\).  Because of the form x\(TraditionalForm\`\^2\) = 4 p y, one can easily see that p = 3.  Thus the focus is the point (0,3) and the directrix is the line 
y = -3 as is seen on the graph.

Now let's have Mathematica demonstrate the definition of a parabola.  What is shown below is that the distance from the focus to a point on the parabola is the same as the perpendicular distance from the point to the directrix.

Do[pt=Graphics[{PointSize[0.04],RGBColor[0,0,1],Point[{t,f[t]}]}];
   line1=Graphics[Line[{{t,f[t]},{t,-3}}]];
   line2=Graphics[Line[{{t,f[t]},{0,3}}]];  
   labels=Graphics[{Text[FontForm["F",{"Courier",12}], 
   {-1,3}],
      	            Text[FontForm["D",{"Courier",12}], 
      	            {-1,-3}],
      	            Text["(x,f(x))",{t+1,f[t]+1}]}];
   disf=N[Sqrt[t^2 + (f[t]-3)^2],3];
   disd=N[(f[t] + 3),3];
   distances=Graphics[{Text[FontForm["Distance to F = ",
   {"Courier",12}],{2,-5},{1,0}],
     	          Text[FontForm[disf,{"Courier",12}],
     	           {2,-5}, {-1,0}],
     	          Text[FontForm["Distance to d = ",
     	          {"Courier",12}], {2,5}, {1,0}],
     	          Text[FontForm[disd,{"Courier",12}], 
     	          {2,5}, {-1,0}]}];
   Show[p1,line1,line2,focus,pt,labels,distances,
   DisplayFunction->$DisplayFunction],
   {t,-6,6,2}]

Select the above cells and press command-y to animate.