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Chapter 15: Problem 1440

If \(\mathrm{AB}\) is a double ordinates of the hyperbola \(\left(\mathrm{x}^{2} / \mathrm{a}^{2}\right)-\left(\mathrm{y}^{2} / \mathrm{b}^{2}\right)=1\) such that \(\mathrm{OAB}\) 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) \(\mathrm{e}<(1 / \sqrt{3})\) (c) \(\mathrm{e}>(\sqrt{3} / 2)\) (d) \(e>(2 / \sqrt{3})\)

Short Answer

Expert verified
The eccentricity e of the hyperbola satisfies \(1 < e < \frac{2}{\sqrt{3}}\) and \(e > \frac{\sqrt{3}}{2}\).

Step by step solution

01

Find Coordinates of A and B

A and B have the same x-coordinate since AB is a vertical line. The equation of tangents to the hyperbola in parametric form is \(y = mx \pm \frac{b}{a} \sqrt{m^2 - 1}\). Since A and B are symmetrical with respect to the x-axis, we know that the y-coordinate of B is the negative of the y-coordinate of A. So, A and B coordinates are (px, py) and (px, -py), respectively.
02

Express OA and AB using coordinates

The distance between O and A can be found using the distance formula: \(OA = \sqrt{(px)^2 + (py)^2}\). Similarly, we find AB: \(AB = 2py\).
03

Equilateral triangle condition

In an equilateral triangle, all sides are equal. Using this condition, we have \(OA = OB = AB\). Thus, \(OA = AB = 2py\). Therefore, \(\sqrt{(px)^2 + (py)^2} = 2py\).
04

Relation between a and b

From the equilateral triangle condition, \(px = \sqrt{3}py\). Substituting px value in the hyperbola equation, we get \[\frac{b^2}{a^2} = \frac{\sqrt{3}^2 - 1}{\sqrt{3}^2}\]
05

Determine the possible values of e

Substituting the relationship between a and b in the eccentricity equation, we obtain \(e = \sqrt{1 + \frac{2}{3}}\). This simplifies to \(e = \sqrt{\frac{5}{3}}\). Now, we can compare this value to the given options: (a) \(1 < e < \frac{2}{\sqrt{3}}\) => \(1 < \sqrt{\frac{5}{3}}\) => This is true. (b) \(e < \frac{1}{\sqrt{3}}\) => This is false. (c) \(e > \frac{\sqrt{3}}{2}\) => This is true. (d) \(e > \frac{2}{\sqrt{3}}\) => This is false. So the correct options are (a) and (c).

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