Question

1) Let $z_1=2-i,z_2=-2+i,$ find
i) $Re\left(\frac{z_1{z}_2}{{\overline{z}}_1}\right)$

2) Let $z_1=2-i,z_2=-2+i,$ find
ii) $Im\left(\frac{1}{z_1{\overline{z}}_1}\right)$

Answer 1

1) Given data : $z_1=2-i,z_2=-2+i$

Find the value of $\frac{z_1{z}_2}{{\overline{z}}_1}$

$\therefore z_1{z}_2=(2-i)(-2+i)=-4+2i+2i-(i{)}^2$ { $\because i^2=-1$}

$=-4+4i+1=-3+4i$ .......(i)

And ${\overline{z}}_1$ = conjugate of $z_1=2+i$ ...(ii)

From Equation (i) and (ii),

$\frac{z_1{z}_2}{{\overline{z}}_1}=\frac{-3+4i}{2+i}$

Do rationalisation
On multiplying numerator and denominator by $(2-i)$, we obtain

$\frac{z_1{z}_2}{{\overline{z}}_1}=\frac{-3+4i}{2+i}\times \frac{2-i}{2-i}$

$=\frac{-6+3i+8i-4i^2}{(2{)}^2-(i{)}^2}$

$=\frac{-6+11i-4(-1)}{4+1}$

$=\frac{-2+11i}{5}$

$=\frac{-2}{5}+\frac{11i}{5}$

$\frac{z_1{z}_2}{{\overline{z}}_1}=\frac{-2}{5}+\frac{11i}{5}$

Compare real parts

On comparing real parts, we obtain

$Re\left(\frac{z_1{z}_2}{{\overline{z}}_1}\right)=-\frac{2}{5}$

2) Given data : $z_1=2-i,z_2=-2+i$

Find the value of $\frac{1}{z_1{\overline{z}}_1}$

$\therefore {\overline{Z}}_1$ = conjugate of $z_1=2+i$

Now, $\frac{1}{z_1{\overline{z}}_1}=\frac{1}{(2-i)(2+i)}=\frac{1}{(2{)}^2-(i{)}^2}=\frac{1}{4+1}=\frac{1}{5}$

Compare imaginary parts

On comparing imaginary parts, we get

$Im\left(\frac{1}{z_1{\overline{z}}_1}\right)=0$