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LetH:(x^(2))/(a^(2))-(y^(2))/(b^(2))=1, ...

Let`H:(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`, where a gt b gt 0, be a hperbola in the xy-plane whose conjugate axis LM subtends and angle of `60^(@)` at one of its vertices N. Let the area of the triangle LMN be `4sqrt3`.

The correct option is :

A

`PrarrIV, QrarrII, R rarrI,SrarrIII`

B

`PrarrIV,QrarrIII,R rarrI, SrarrII`

C

`PrarrIV,QrarrI,R rarrIII, SrarrII`

D

`PrarrIII, QrarrIV,R rarrII, S rarrI`

Text Solution

Verified by Experts

The correct Answer is:
B

`" Area of " DeltaLMN=4sqrt3`
`(1)/(2)(2b)(sqrt3b)=4sqrt3`
`rArr" "b^(2)=4rArrb=2rArr2b=4`
Here, `(a)/(b)=cot 30^(@)rArra=sqrt3b rArra=2sqrt3`
Now, `b^(2)=a^(2)(e^(2)-1)`
`therefore" "4=12(e^(2)-1)`
`rArr" "e^(2)=1+(1)/(3)=(4)/(3)`
`rArr" "e=(2)/(sqrt3)`
`"Distance between foci"=2ae=2xx2sqrt3xx(2)/(sqrt3)=8`
`"Lenght of latus rectum "=(2b^(2))/(a)=(2xx4)/(2sqrt3)=(4)/(sqrt3)`
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