On a rectangular hy perbola x^2-y^2= a ^2, a | vedclass

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Question

(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

Vedclass · Accepted Answer

$(2,4]$

Vedclass · Answer

(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, $k \in(2,4]$.

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