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  • Let    be the latus rectum of a hyperbola and <img alt="\mathrm{S_1}" sr

Let  \mathrm{L L_1}  be the latus rectum of a hyperbola and \mathrm{S_1} is the other focus.

If triangle \mathrm{L L_1S_1} is an equilateral triangle, then the eccentricity is

\mathrm{\sqrt{\frac{24}{\lambda}}}  . The value of \mathrm{\lambda} is

Option: 1

8


Option: 2

7


Option: 3

4


Option: 4

9


Answers (1)

best_answer

\mathrm{S_1 S=2 a e}

\mathrm{L S=\frac{b^2}{a}=a\left(e^2-1\right)}                         

\mathrm{\begin{aligned} & \tan 30^{\circ}=\frac{L S}{S_1 S}=\frac{e^2-1}{2 e}=\frac{1}{\sqrt{3}} \\ & \Rightarrow e^2-\frac{2}{\sqrt{3}} e-1=0 \\ & \Rightarrow e=\sqrt{3}=\sqrt{\frac{24}{\lambda}} \Rightarrow \lambda=8 \end{aligned}}

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