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Question

Let a,bR and a2+b20. Suppose S={zC:z=1a+ibt,tR,t0}, where i=1. If z=x+iy and zS, then (x,y) lies on

A
the circle with radius 12a and centre (12a,0)for a>0,b0
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B
the circle with radius 12a and centre (12a,0)for a<0,b0
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C
the x-axis for a0,b=0
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D
the y-axis for a=0,b0
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Solution

The correct options are
A the circle with radius 12a and centre (12a,0)for a>0,b0
C the x-axis for a0,b=0
D the y-axis for a=0,b0
z=1a+ibtz=aibta2+b2t2=x+iyx=aa2+b2t2 and y=bta2+b2t2

So assuming a0 and b0 x2+y2=1a2+b2t2=xax2xa+y2=0(x12a)2+y2=(12a)2

If a0,b=0, then (x,y)=(1a,0)

If a=0,b0, then (x,y)=(0,1bt)

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