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

Options

A. `e=sqrt(386)/12`
B. `e=sqrt(386)/13`
C. `LR=121//6`
D. `LR=121//3`

Answer

A, C

Explanation

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)`