KVPY SX SB Exam 2018 Question Paper

Question : 5 of 120

Marks: +1, -0

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 triangle ABC is equilateral. If the side-length of the triangle ABC is ka, then k lies in the interval

(0, 2]
(2, 4]
(4, 6]
(6, 8]

Validate

Solution:

\AB^2 = \BC^2

(2a \tan θ)^2 = a^2(\sec θ + 1)^2 + a^2(\tan^2 θ)

4\ tan^2θ = \sec^2 θ + 1 + 2 \sec θ + \tan^2 θ

\tan^2 θ = 1 + \sec θ

\sec^2 θ- \sec θ -2 = 0

(\sec θ - 2) (\sec θ + 1) = 0

\sec θ = +2, \sec θ = –1

(not possible)

\tan θ = √3

length

= 2a √3

⇒ k = 2√ 3

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