Question

If \( P \) is a double ordinate of hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] such that \( OPQ \) is an equilateral triangle, \( O \) being the centre of the hyperbola, then the eccentricity \( e \) of the hyperbola satisfies

• A. \( 1<e<\frac{2}{\sqrt{3}} \)
• B. \( e = \frac{2}{\sqrt{3}} \)
• C. \( e = \frac{\sqrt{3}}{2} \)
• D. \( e>\frac{2}{\sqrt{3}} \)

Hint:

For hyperbolas with double ordinates, use the properties of the eccentricity and the geometry of the curve to solve for the required values.

Answer 1

The Correct Option is A

Step 1: Using properties of hyperbolas.
For a double ordinate of a hyperbola and an equilateral triangle formed with the center of the hyperbola, the eccentricity satisfies the condition \( 1<e<\frac{2}{\sqrt{3}} \).

Step 2: Conclusion.
The correct range for the eccentricity is \( 1<e<\frac{2}{\sqrt{3}} \), corresponding to option (1).