A) Circle of radius \[\frac{\sqrt{5}}{2}\]
B) Circle of radius \[\frac{5}{4}\]
C) Straight line
D) Parabola
Correct Answer: A
Solution :
Given that Im\[\left( \frac{z+i}{z+2} \right)\] Let \[z=x+iy\]Þ \[\frac{x+iy+i}{x+iy+2}\]=\[\frac{x+i\,(y+1)}{(x+2)+iy}\] \[=\frac{[x+i(y+1)][(x+2)-iy]}{[(x+2)+iy][(x+2)-iy]}\] \[=\left[ \frac{{{x}^{2}}+2x+{{y}^{2}}+y}{{{(x+2)}^{2}}+{{y}^{2}}} \right]+i\,\left[ \frac{(y+1)(x+2)-xy}{{{(x+2)}^{2}}+{{y}^{2}}} \right]\] If it is purely imaginary then real part must be equal to zero. Þ \[\frac{{{x}^{2}}+{{y}^{2}}+2x+y}{{{(x+2)}^{2}}+{{y}^{2}}}=0\]Þ \[{{x}^{2}}+{{y}^{2}}+2x+y=0\] Which is a circle and its radius is given by \[\sqrt{{{g}^{2}}+{{f}^{2}}-c}=\sqrt{1+\frac{1}{4}-0}=\frac{\sqrt{5}}{2}\] Therefore Argand diagram is circle of radius\[\frac{\sqrt{5}}{2}\].
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