WBJEE · Maths · Hyperbola
\(P Q\) is a double ordinate of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) such that \(\triangle O P Q\) is an equilateral triangle, \(O\) being the centre of the hyperbola. Then the eccentricity e of the hyperbola satisfies
- A \(1 < \mathrm{e} < 2 / \sqrt{3}\)
- B \(\mathrm{e}=2 / \sqrt{3}\)
- C \(\mathrm{e}=2 \sqrt{3}\)
- D \(\mathrm{e} > 2 / \sqrt{3}\)
Answer & Solution
Correct Answer
(D) \(\mathrm{e} > 2 / \sqrt{3}\)
Step-by-step Solution
Detailed explanation
\(\tan 30^{\circ}=\frac{b \tan \theta}{a \sec \theta}\) \(\Rightarrow \frac{b}{a}=\frac{1}{\sin \theta \sqrt{3}}\) \(e=\sqrt{1+\frac{1}{3 \sin ^2 \theta}} > \sqrt{1+\frac{1}{3}}\) \(\Rightarrow e > \frac{2}{\sqrt{3}}\left(0
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