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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
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Solution
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.
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