Question

The difference of the focal distances of any point on the hyperbola is constant.

Answer 1

Let P(x, y) be any point on the hyperbola whose centre is the origin C foci are S, S' directries are ZM and Z'M' as shown in fig. 

Let PN, PM, PM' be the perpendiculars drawn from P upon x–axis and the two directrices respectively. 

Now SP = e(PM) = e(NZ) = e(CN – CZ). 

∴ SP = e(x - a/e) = ex - a.

and S'P = e(PM') = e(NZ') = e(CN + CZ')

= e(x + a/e) = ex + a

∴ S'P – SP = 2a. 

By the above theorem, the hyperbola is sometimes defined as the locus of a point, the difference of whose distances from two fixed points is constant.