Rectangular Hyperbola Distances

hypinv.html

Exploration

Show that the distance of any point on a rectangular hyperbola from its center varies inversely as the perpendicular distance from its polar to the center.

Approach

Construct a generic point on a rectangular hyperbola and compare the appropriate distances.

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 generic point on a rectangular hyperbola.

Clear[a,t];
h1=Hyperbola2D[{0,0},a,a,0];
p1=Point2D[a*Cosh[t],a*Sinh[t]]

Compute the distances.

p0=Point2D[0,0];
{D1,D2}=
   {Distance2D[p0,p1],
    Distance2D[p0,l1=Line2D[p1,h1]]} //Simplify

Use a trigonometric identity.

1/Sech[2t] //Simplify

Therefore, since is a constant, varies inversely as .

Clear[E1];
D1*D2 //. {
   Sqrt[a^2*E1_]->a*Sqrt[E1],
   Sqrt[Cosh[E1_]]*Sqrt[Sech[E1_]]->1}
   

Discussion

This is a plot of a numerical example of the geometric objects.

Sketch2D[{h1,p1,p0,l1} /. {a->1,t->0.5}]

Graphics saved as "hypinv01.eps".