Question

Show that the absolute value of difference of the focal distances of any point P on the hyperbola is the length of its transverse axis

Answer 1

Let P be a point on the hyperbola.

Definition of conic

`"SP"/"PM"` = e

`"S'P"/"PM'"` = e

SP = e(PM) ……..(1)

S’P = e (PM’) ……….(2)

(2) – (1)

⇒ S’P – SP = e PM’- e PM

= e(PM’ – PM)

= e MM’

= e ZZ’  .......[∵ MM’ = ZZ’ = `(2"a")/"e"`]

= `"e"((2"a")/"e")`

S’P – SP = 2a  .......(constant)

= length of the transverse axis.