The points on the curve xy2=1 which are nearest to the origin, are
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)=0⇒−4t5+2t2√1t4+t2=0−4t5+2t=0⇒t6=2⇒t=±21/6⇒t2=21/3⇒1t2=121/3=(2)−1/3
So, the points on the curve nearest to origin are (2−1/3,±21/6) or ((12)1/3,±(12)−1/6).
So, option A is correct.