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If (5,12)a n d(24 ,7) are the foci of a ...

If `(5,12)a n d(24 ,7)` are the foci of a hyperbola passing through the origin, then (a)`e=(sqrt(386))/(12)` (b) `e=(sqrt(386))/(13)` (c)`L R=(121)/6` (d) `L R=(121)/3`

A

`e=sqrt(386)/12`

B

`e=sqrt(386)/13`

C

`LR=121//6`

D

`LR=121//3`

Text Solution

Verified by Experts

The correct Answer is:
A, C
We have foci `S_(1)(5, 12) and S_(2)(24, 7)`.
Hyperbola passes through P(0, 0).
`therefore" "|PS_(1)-PS_(2)|=2a`
`2a=K`
`"or "2a=|sqrt((24-0)^(2)+(7-0)^(2))-sqrt(12^(2)+5^(2))|=12`
`therefore" "a=6`
`S_(1)S_(2)=2ae=sqrt((24-5)^(2)+(12-7)^(2))=sqrt(386)`
`therefore" "e=(sqrt(386))/(12)`
`LR = (2b^(2))/(a)=(2a^(2)(e^(2)-1))/(a)`
`=2xx6((386)/(144)-1)=(121)/(6)`
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