Question

If $z=x+iy$ and $w=\frac{1-zi}{z-i},|w|=1$, then find the locus of z.

• A. z lies on the imaginary axis.
• B. z lies only on positive real axis.
• C. z lies only on negative real axis.
• D. z lies on the real axis.

Answer 1

The correct option is D z lies on the real axis.

We have,
$\left|w\right|=1⟹\left|\frac{1-iz}{z-i}\right|=1$
or $\frac{|1-iz|}{|z-i|}=1$
or $|1-iz|=|z-i|$ (1)
or $|-i(x+iy\left)\right|=|x+iy-i|$, where $z=x+iy$
or $|1+y-ix|=|x+i(y-1\left)\right|$
or $\sqrt{(1+y{)}^2+(-x{)}^2}=\sqrt{x^2+(y-1{)}^1}$
or $(1+y{)}^2+x^2+x^2+(y-1{)}^2$
or $y=0$
$⟹z=x+i0=x,$ which is purely real
Ans: D