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  • If PQ is a double ordinate of the hyperbola   <img alt="\mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cfrac%7Bx%5E2%7D%7Ba

If PQ is a double ordinate of the hyperbola   \mathrm{\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 find minimum value of eccentricity (e) of the

hyperbola.

Option: 1

\mathrm{\frac{4}{3}}


Option: 2

\mathrm{\frac{2}{\sqrt{3}}}


Option: 3

\mathrm{\frac{3}{2}}


Option: 4

None of these


Answers (1)

best_answer

Let the hyperbola be  \mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}  and any double ordinate

PQ be such that  \mathrm{P \equiv(a \sec \theta, b \tan \theta)}

\mathrm{\therefore Q \equiv(a \sec \theta,-b \tan \theta)}

According to the question, triangle OPQ is equilateral

\mathrm{\begin{aligned} & \therefore \quad \tan 30^{\circ}=\frac{b \tan \theta}{a \sec \theta} \\ & \Rightarrow \quad 3 \frac{b^2}{a^2}=\operatorname{cosec}^2 \theta \\ & \Rightarrow 3\left(e^2-1\right)=\operatorname{cosec}^2 \theta \\ & \text { Now, } \operatorname{cosec}^2 \theta \geq 1 \\ & \Rightarrow 3\left(e^2-1\right) \geq 1 \Rightarrow e^2 \geq \frac{4}{3} \Rightarrow e \geq \frac{2}{\sqrt{3}} \end{aligned}}

Posted by

Ajit Kumar Dubey

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