Eccentricities of Conjugate Hyperbolas

hypeccen.html

Exploration

Show that if and are the eccentricities of a hyperbola and its conjugate, then

.

Approach

Create the hyperbolas, compute their eccentricities and verify the relationship.

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

Create a hyperbola and its conjugate.

Clear[a,b];
h1=Hyperbola2D[{0,0},a,b,0];
h2=Hyperbola2D[h1,Conjugate2D];

Compute the eccentricities of the hyperbolas.

{e1=Eccentricity2D[h1],
e2=Eccentricity2D[h2]}

Verify the relationship.

1/e1^2+1/e2^2 //Simplify