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If P Q is a double ordinate of the hyper...

If `P Q` is a double ordinate of the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` such that `O P Q` is an equilateral triangle, `O` being the center of the hyperbola, then find the range of the eccentricity `e` of the hyperbola.

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Let the hyperbola be `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`
and any double ordinate PQ be such that `P-=(a sec theta,b tan theta)`.
`therefore" "Q-=(a sec theta, -b tan theta)`
According to the question, triangle OPQ is equilateral.
`therefore" "tan30^(@)=(b tan theta)/(a sec theta)`
`rArr" "3(b^(2))/(a^(2))="cosec"^(2)theta`
`rArr" "3(e^(2)-1)="cosec"^(2)theta`
Now, `"cosec"^(2)thetage1`
`rArr" "3(e^(2)-1)ge1`
`rArr" "e^(2)ge(4)/(3)`
`rArr" "ege(2)/(sqrt3)`
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