A) A straight line
B) A circle
C) An ellipse
D) A pair of straight lines
Correct Answer: A
Solution :
\[|iz-1|+|z-i|\,=2\] \[\Rightarrow \] \[|i(x+iy)-1|\,+|x+iy-i|\,=\,2\] \[\Rightarrow \] \[|-(y+1)+ix|\,+|x+i(y-1)|=2\] \[\Rightarrow \] \[\sqrt{{{(-(y+1))}^{2}}+{{x}^{2}}}+\sqrt{{{x}^{2}}+{{(y-1)}^{2}}}=2\] \[\Rightarrow \] \[\sqrt{{{(y+1)}^{2}}+{{x}^{2}}}=\,2-\sqrt{{{x}^{2}}+{{(y-1)}^{2}}}\] \[\Rightarrow \] \[{{y}^{2}}+1+2y+{{x}^{2}}=\,4+{{x}^{2}}+{{y}^{2}}+1-2y-4\sqrt{{{x}^{2}}+{{(y-1)}^{2}}}\] \[\Rightarrow \]\[4y=4-4\sqrt{{{x}^{2}}+{{(y-1)}^{2}}}\]\[\Rightarrow \]\[y=1-\sqrt{{{x}^{2}}+{{(y-1)}^{2}}}\] \[\Rightarrow \] \[{{x}^{2}}+{{(y-1)}^{2}}={{(1-y)}^{2}}\] \[\Rightarrow \] \[{{x}^{2}}+{{y}^{2}}+1-2y=\,1+{{y}^{2}}-2y\]\[\Rightarrow \]\[{{x}^{2}}=0\Rightarrow x=0\] i.e. equation of straight line.
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