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Question

If PQ is a double ordinate of the hyperbola x2a2y2b2=1 such that OPQ is an equilateral triangle, O being the centre of the hyperbola, then the eccentricity e of the hyperbola satisfies

A
1<e<23
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B
e>23
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C
e=23
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D
e>43
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Solution

The correct option is B e>23
Let the coordinates of P be (h,k) and coordinates of Q be (h,k).
O(0,0) is the centre.
OPQ is equilateral.
OP2=OQ2=PQ2h2+k2=4k2h2=3k2

Also, (h,k) lies on the hyperbola.
h2a2k2b2=13k2a2k2b2=1k2=a2b23b2a2>03b2a2>0b2a2>13

Now, e2=1+b2a2
e2>43
e>23

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