Polar Equation of a Parabola

polarpb.html

Exploration

Show that the polar equation of a parabola opening to the right with vertex at (0,0) is given by

where f is the focal length of the parabola.

Approach

Create the parabola in rectangular coordinates. Convert the equation to polar coordinates.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Construct the quadratic representing the parabola.

Clear[f];
Q1=Quadratic2D[Parabola2D[{0,0},f,0]]

Convert the equation from rectangular coordinates to polar coordinates.

Clear[x,y,r,theta];
eq1=Equation2D[Q1,{x,y}] /.
    {x->r*Cos[theta],y->r*Sin[theta]}

Solve for r to get the desired form of the equation.

Solve[eq1,r]

The trigonometric identity completes the demonstration.

4f*Cos[theta]/Sin[theta]^2 //Simplify