The region of the z-plane for which ∣∣∣z−az+¯a∣∣∣=1(Rea≠0) is X-axis.
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∣∣∣z−az+¯a∣∣∣=1⇒|z−a|2=|z+¯a|2 If z=(x,y),a=(p,q) then ¯a=(p,q) ∴(x−p)2+(y−q)2=(x+p)2+(y−q)2 ∴(x+p)2−(x−p)2=0or 4px=0∴x=0 i.e.y-axis as p≠0 This is the equation of y-axis.
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