Question

Let PQ be a double ordinate of hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}$ = 1. If O be the centre of the hyperbola and OPQ is an equilateral triangle then the eccentricity e is –

• A. >$\sqrt{3}$
• B. > 2
• C. > 2/$\sqrt{3}$
• D. None of these

Answer 1

Answer: (C)

> 2/$\sqrt{3}$

Answer 2

P be (a, b) then PQ = 2b

OP = $\sqrt{\alpha^{2} + \beta^{2}}$

Since OPQ is an equilateral triangle

OP = PQ

a² + b² = 4 b² = a² = 3b² Ž a = ±$\sqrt{3}$b

(a, b) is situated on the hyperbola

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}$ = 1 $\frac{\alpha^{2}}{a^{2}} - \frac{\beta^{2}}{b^{2}}$ = 1

$\frac{3\beta^{2}}{a^{2}} - \frac{\beta^{2}}{b^{2}}$ = 1 Ž $\frac{3}{a^{2}}$ – $\frac{1}{b^{2}}$= $\frac{1}{\beta^{2}}$ > 0

$\frac{b^{2}}{a^{2}}$ > 1/3 Ž e² – 1 > 1/3

e² > 4/3 Ž e > 2/$\sqrt{3}$