Question

The value of $\left(\text{Z+3)}\right(\overline{\text{Z}}+3)$ is equivalent to :

Answer 1

We know that $\text{z=x+iy}$ and $\overline{z}=\text{x-iy}$

Now we calculate $\text{z+3=x+iy+3 ,}\overline{z}\text{=x-iy+3}$

$\text{(x+iy+3)(x-iy+3)=(x+3+iy)(x+3-iy)}$ (now we use ${\text{(a-b)(a+b)=a}}^2{\text{-b}}^2$)

Here we assume $\text{x+3=a and iy=b}$ so

$$\begin{gathered} {\text{(x+3)}}^2{\text{-(iy)}}^2{\text{=x}}^2{\text{+3}}^2{\text{+2.3.x-(i}}^2{\text{y}}^2\text{)} \\ ={\text{x}}^2{\text{+9+6x-(-y}}^2{\text{) (here i}}^2\text{=-1)} \\ ={\text{x}}^2{\text{+9+6x+y}}^2 \end{gathered}$$

We know that $\left|z\right|=\sqrt{{\text{x}}^2{\text{+y}}^2}$ and here we use it

${\text{x}}^2{\text{+6x+9+y}}^2\text{=}$$({\text{x+3)}}^2{\text{+y}}^2\text{=}{\left|\text{x+3+iy}\right|}^2\text{=}{\left|\text{z+3}\right|}^2$