Polar Equation of a Hyperbola

polarhyp.html

Exploration

Show that the polar equation of a hyperbola with a horizontal transverse axis and centered at (0,0) is given by

.

Approach

Create the hyperbola in rectangular coordinates and 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

Define a quadratic representing the hyperbola.

Clear[a,b];
Q1=Quadratic2D[Hyperbola2D[{0,0},a,b,0]]

Convert from rectangular 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 put the equation into the desired form.

ans=Solve[eq1,r]

Multiply the fraction by to get the desired form.

Clear[E1,E2];
Last[ans] /. {I*E1_/Sqrt[E2_]->E1/Sqrt[-E2]}