Question

If the angle between the asymptotes of the hyperbola \( x^2 - ky^2 = 3 \) is \( \frac{\pi}{3} \) and e is its eccentricity, then the pole of the line \( x + y - 1 = 0 \) w.r.t. this hyperbola is:

• A. \( \left( k, \frac{\sqrt{3}e}{2} \right) \)
• B. \( \left( -k, \frac{\sqrt{3}e}{2} \right) \)
• C. \( \left( -k, -\frac{\sqrt{3}e}{2} \right) \)
• D. \( \left( k, -\frac{\sqrt{3}e}{2} \right) \)

Hint:

For hyperbola asymptote and pole problems, apply trigonometric identities to solve for parameters first.

Answer 1

The Correct Option is D

Step 1: Finding the hyperbola parameters The standard form of a hyperbola is: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] The angle between asymptotes is given by: \[ \theta = 2 \tan^{-1} \left( \frac{b}{a} \right) \] Using \( \theta = \frac{\pi}{3} \), we find \( a, b \), and compute eccentricity \( e \). Step 2: Finding the pole The pole equation is determined by using the relation for pole with respect to hyperbola.