Question

Let $z=x+iy$ be a complex number. Then which of the following statements is (are) TRUE ?

• A. If $\frac{z^2}{z+1}\in R,$ then $z$ either lies on circle $x^2+y^2+2x=0$ or on real axis excluding the point $(-1,0).$
• B. If $\left|z\right|=4\sqrt{5},$ then the points representing the complex number $3+\sqrt{5}z$ lie on a circle of radius $20$ units.
• C. If $z=(2+a)+i\sqrt{3-a^2},$ where $a\in R$ and $a^2<3,$ then locus of $z$ for different value of $a$ is a semi-circle with centre $(2,0).$
• D. If $\omega$ is a complex number such that $\omega =\frac{1-iz}{z-i}$ and $|\omega |=1,$ then $z$ lies on the real axis.

Answer 1

Answer: (A), (B), (C), (D)

If $\omega$ is a complex number such that $\omega =\frac{1-iz}{z-i}$ and $|\omega |=1,$ then $z$ lies on the real axis.

$\frac{z^2}{z+1}=\overline{\frac{z^2}{z+1}}$
$\Rightarrow \frac{z^2}{z+1}=\frac{{\overline{z}}^2}{\overline{z}+1}$
$\Rightarrow z^2\overline{z}+z^2=z{\overline{z}}^2+{\overline{z}}^2$
$\Rightarrow z\overline{z}(z-\overline{z})+(z+\overline{z})(z-\overline{z})=0$
$\Rightarrow z\overline{z}+z+\overline{z}=0$ or $z=\overline{z}$
$\Rightarrow x^2+y^2+2x=0$ or $y=0$

$\left|z\right|=4\sqrt{5}$
Let $\omega =3+\sqrt{5}z$
$|\omega -3|=\sqrt{5}\times 4\sqrt{5}=20$

$z=(2+a)+i\sqrt{3-a^2}=x+iy$
$\Rightarrow x=2+a$ and $y=\sqrt{3-a^2}\ge 0$
$\Rightarrow (x-2{)}^2+y^2=3;y\ge 0$

$|\omega |=1$
$\Rightarrow \left|\frac{1-iz}{z-i}\right|=1$
$\Rightarrow |1-iz|=|z-i|$
$\Rightarrow |-i||z+i|=|z-i|$
$\Rightarrow |z+i|=|z-i|$
This means $z$ is equidistant from $(0,-1)$ and $(0,1)$
So, $z$ lies on the perpendicular bisector of $(0,-1)$ and $(0,1)$ i.e., $x-$axis.