Question

If the imaginary part of the expression $\frac{z-1}{e^{i\theta }}+\frac{e^{i\theta }}{z-1}$ zero, then determine the locus of the point z.

• A. Above represents a straight line passing through the point (0, 1).
• B. Above represents a circle with center (1,0) and radius 1.
• C. Either A or B
• D. above represents a circle with center (0,1) and radius 1

Answer 1

Answer: (C)

Either A or B

Let $z=x+iy$
Hence
$z-1=(x-1)+iy=r{e}^{i\alpha }$.
Hence the above expression reduces to
$r{e}^{i(\alpha -\theta )}+\frac{1}{r}.e^{i(\theta -\alpha )}$
Considering the imaginary part, we get
$rsin(\alpha -\theta )+\frac{1}{r}sin(\theta -\alpha )$
$rsin(\alpha -\theta )-\frac{1}{r}sin(\alpha -\theta )$
$=(r-\frac{1}{r})sin(\alpha -\theta )$
$=0$
Hence Either
$r-\frac{1}{r}=0$
$r^2=1$
$(x-1{)}^2+y^2=1$ ... Equation of circle centered at (1,0).
Or
$sin(\alpha -\theta )=0$
Or
$\alpha -\theta =0$
Or
$\alpha =\theta$
Or
$tan\alpha =tan\theta$
Or
$\frac{y}{x-1}=tan\theta =m$ where m is the slope of the line.
Hence
$y=m(x-1)$... Equation of a line passing through (1,0).