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On a rectangular hy perbola $x^2-y^2= a ^2, a >0$, three points $A, B, C$ are taken as follows: $A=(-a, 0) ; B$ and $C$ are placed symmetrically with respect to the $X$-axis on the branch of the hyperbola not containing $A$. Suppose that the $\triangle A B C$ is equilateral. If the side length of the $\triangle A B C$ is $k a$, then $k$ lies in the interval
$(0,2]$
$(2,4]$
$(4,6]$
$(6,8]$
Solution
(b)
We have rectangular hyperbola
$x^2-y^2=a^2$
Given $A B C$ is an equilateral triangle.
$A B =B C=A C$
$A B^2 =B C^2$
$a^2(\sec \theta+1)^2+a^2 \tan ^2 \theta=4 a^2 \tan ^2 \theta$
$\begin{aligned}(\sec \theta+1)^2 &=3 \tan ^2 \theta \\(\sec \theta+1)^2 &=3\left(\sec ^2 \theta-1\right) \\(\sec \theta+1)^2 &=3(\sec \theta+1)(\sec \theta-1) \end{aligned}$
$\sec \theta+1=3 \sec \theta-3$
$\sec \theta=2$
$\theta=60^{\circ}$
$\because$ Side $B C=2 a \tan \theta$
$=2 a \tan 6 \theta^{\circ}=2 a \sqrt{3}$
But side of triangle is $k a$.
$k a =2 a \sqrt{3}$
$k =2 \sqrt{3}$
Hence, $k \in(2,4]$.
