(notes by R. Bigoni)

Let γ be a hyperbola with semimajor axis *a* and focuses *F'* and
*F''* and let *P* be a point of γ. It can be demonstrated that the absolute
value of the difference between the distances from *P* to *F'* and from *P*
to *F''* is equal to the measure of the major axis *2a*.

With reference to the figure, let 2*c* be the measure of the segment *F'F''*
and let *P* be a point on the right branch, that which
has the vertex closer to the focus *F'*. Let ρ be the measure of the segment
*F'P* and *r* the measure of the segment *F''P*.

By applying the cosine rule to the triangle *F'F''P* we obtain

Since *c = a e* we have

From the polar equation of the conic (2.2) we obtain

so the (3) can be rewritten as

Finally, using the expression of *a* in terms
of *e* and *l*

we have

Since the symmetry of the hyperbola, if the point *P* belongs to the other branch
of the curve, that which has vertex closer to the focus *F''*, we can obtain