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Question

The line x + y = 1 meets x-axis at A and y axis at B, P is the mid point of AB, P1 is the foot of the perpendicular from P to OA; M1 is that of P1 from OP; P2 is that of M1 from OA; M2 is that of P2 from OP; P3 is that of M2 from OA and so on. If Pn denotes the nth foot of the perpendicular on OA from Mn1, then OPn is equal to
  1. 12n
  2. 12n
  3. 2n1
  4. 2n+1


Solution

The correct option is B 12n
Equation of AB is x + y = 1, then coordinates of A and B are (1, 0) and (0, 1) respectively.
coordinates of P is (12,12)
PP1 is perpendicular to OA
Equation OP is y = x
OP1=PP1=12

We have (OMn1)2=(OPn)2+(PnMn1)2
=2(OPn)2          {y=x)
=2(OPn)2=2α2n        {where we let OPn=αn}
We also have
(OPn1)2=(OMn1)2+(Pn1Mn1)2α2n1=2α2n+12α2n112αn1=2α2nαn=12αn1=122αn2=123αn3=...=12n112=12n

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