Let the tranverse axis be along the x-axis and the conjugate axis be
along the y-axis, with the center C at (0,0), the origin.

Let both axes be 2, so that both the semi-tranverse axis, "a", and 
semi-conjugate axis, "b", are 1 each.   Then the equation of the 
hyperbola, which is , becomes simply .
In a hyperbola, , so , therefore
, where "c" is the distance from the center to the focus.
Therefore S and S' are the points (,0).  

[Do not confuse the center "C(0,0)" with the value of "c", the distance 
from the center to each focus.]

Let P(x,y) be any arbitrary point on the hyperbola:
The blue line is CP:

The graph is:

 

Using the distance formula to find SP and SP' in terms of x:



Since the equation of the hyperbola is , then ,
so substituting we get:





Similarly,



As before, since the equation of the hyperbola is ,
then , so substituting we get:





So 




      
Multiplying under the radicals:





Next we use the distance formuls to find CP where C is the origin (0,0).



Since the equation of the hyperbola is , then ,
so substituting we get



so 

Therefore , because both equal  

Edwin