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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 from P1 to OP; P2 is that from M1 to OA; M2 is that from P2 to OP; P3 is that from M2 to OA and so on. If Pn denotes the nth foot of the perpendicular on OA from Mn1, then OPn =


A

12

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B

12n

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C

12n2

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D

12

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Solution

The correct option is B

12n




x+y=1 meets x - axis at A(1,0) and y - axis at B(0,1)
The coordinates of P are (12,12) and PP1 is perpendicular to OA.
OP1=P1P=12
Equation of line OP is y=x.
We have (OMn1)2=(OPn)2+(PnMn1)2=2(OPn)2=2p2n (say)
Also, (OPn1)2=(OMn1)2+(Pn1Mn1)2
=2p2n+12p2n1 p2n=14p2n1 pn=12pn1 OPn=pn=12pn1=122pn2==12n1p1=12n


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