Let a,b∈R and a2+b2≠0. Suppose S={z∈C:z=1a+ibt,t∈R,t≠0}, where i=√−1. If z=x+iy and z∈S, then (x,y) lies on
A
the circle with radius 12a and centre (12a,0)for a>0,b≠0
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B
the circle with radius −12a and centre (−12a,0)for a<0,b≠0
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C
the x-axis for a≠0,b=0
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D
the y-axis for a=0,b≠0
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Solution
The correct options are A the circle with radius 12a and centre (12a,0)for a>0,b≠0 C the x-axis for a≠0,b=0 D the y-axis for a=0,b≠0 z=1a+ibt⇒z=a−ibta2+b2t2=x+iy∴x=aa2+b2t2 and y=−bta2+b2t2
So assuming a≠0 and b≠0x2+y2=1a2+b2t2=xa⇒x2−xa+y2=0⇒(x−12a)2+y2=(12a)2