A) a straight line parallel to the .x-axis
B) a circle of radius 1 and centre (1, 0)
C) a parabola
D) None of the above
Correct Answer: B
Solution :
Let\[z-1=r{{e}^{i\alpha }}\] \[\Rightarrow \]\[(x-1)+iy=r(\cos \alpha +i\sin \alpha )\] \[\Rightarrow \]\[{{(x-1)}^{2}}+{{y}^{2}}={{r}^{2}}\]and \[\tan \alpha =\frac{y}{x-1}\] Now,\[\frac{z-1}{{{e}^{i\theta }}}+\frac{{{e}^{i\theta }}}{z-1}=r{{e}^{i(\alpha -\theta )}}+\frac{1}{r}{{e}^{-i(\alpha -\theta )}}\] Since, it is real. \[\therefore \]\[r\sin (\alpha -\theta )-\frac{1}{r}\sin (\alpha -\theta )=0\] \[\Rightarrow \]\[r-\frac{1}{r}=0\]\[\Rightarrow \]\[{{r}^{2}}=1\]\[\Rightarrow \]\[{{(x-1)}^{2}}+{{y}^{2}}=1\] Which is a circle with centre (1,0) and radius 1.
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