On a rectangular hyperbola 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 Δ A B C is equilateral. If the side length of the Δ A B C is k a , then k lies in the interval

Step-by-step solution

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 θ + 1 2 + a 2 tan 2 θ = 4 a 2 tan 2 θ ⇒ sec θ + 1 2 = 3 tan 2 θ ⇒ sec θ + 1 2 = 3 sec 2 θ - 1 ⇒ sec θ + 1 2 = 3 sec θ + 1 sec θ - 1 ⇒ sec θ + 1 = 3 sec θ - 3 ⇒ sec θ = 2 ⇒ θ = 60 ° ∵ Side B C = 2 a tan θ = 2 a tan 60 ° = 2 a 3 But side of triangle is k a . ∴ k a = 2 a 3 k = 2 3 Hence, k ∈ ( 2 , 4 ]