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Question

The points on the curve xy2=1 which are nearest to the origin, are

A
[(12)1/3,±(12)1/6]
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B
[(12)1/3,21/6]
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C
[21/3,±(12)1/6]
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D
None of these
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Solution

The correct option is A [(12)1/3,±(12)1/6]

Any point on the curve xy2=1 is of form P(1t2,t)

Let distance of P from the origin =a

a=(1t2)2+t2a=1t4+t2

At minimum distance dadt=0

ddt(1t4+t2)=04t5+2t21t4+t2=04t5+2t=0t6=2t=±21/6t2=21/31t2=121/3=(2)1/3

So, the points on the curve nearest to origin are (21/3,±21/6) or ((12)1/3,±(12)1/6).

So, option A is correct.


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