A) hyperbola
B) ellipse
C) circle
D) straight line
Correct Answer: D
Solution :
Given, \[z=x+iy\]and \[\left| \frac{z-1}{z+2i} \right|=1\] \[\Rightarrow \] \[\left| \frac{(x+iy)-1}{(x+iy)+2i} \right|=1\] \[\Rightarrow \] \[|(x-1)+iy|=|x+(y+2)i|\] Squaring on both sides, \[|(x-1)+iy{{|}^{2}}=|x+(y+2)i{{|}^{2}}\] \[\Rightarrow \]\[{{(x-1)}^{2}}+{{y}^{2}}={{x}^{2}}+{{(y+2)}^{2}}\] \[\Rightarrow \] \[{{x}^{2}}+{{y}^{2}}+1-2x={{x}^{2}}+{{y}^{2}}+4+4y\] \[\Rightarrow \]\[2x+24+3=0,\] which represents a straight line.
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