Question

The value of $\frac{z{\left|\overline{z}\right|}^2}{\overline{z}\left|z^2\right|}$ is(if complex number $z$ has amplitude $\frac{\pi }{3}$)

• A. $\frac{1}{2}+\frac{\sqrt{3}}{2}i$
• B. $-\frac{1}{2}-\frac{\sqrt{3}}{2}i$
• C. $\frac{1}{2}-\frac{\sqrt{3}}{2}i$
• D. $-\frac{1}{2}+\frac{\sqrt{3}}{2}i$

Answer 1

The correct option is D $-\frac{1}{2}+\frac{\sqrt{3}}{2}i$

Let $z=x+iy$ then, $\overline{z}=x-iy$

$|z|=|\overline{z}|=\sqrt{x^2+y^2}$

$\Rightarrow z^2=(x+iy{)}^2=x^2+i^2{y}^2+2xyi$

$\Rightarrow z^2=(x+iy{)}^2=x^2-y^2+i2xy$

$\Rightarrow |z^2|=\sqrt{(x^2-y^2{)}^2+4x^2{y}^2}$

$\Rightarrow |z^2|=\sqrt{(x^2+y^2{)}^2}$

$\Rightarrow |z^2|=x^2+y^2$

and given that ${\tan }^{-1}\left(\frac{y}{x}\right)=\frac{\pi }{3}$

$\Rightarrow \frac{y}{x}=\tan \left(\frac{\pi }{3}\right)$

$\Rightarrow \frac{y}{x}=\sqrt{3}$

$\Rightarrow y^2=3x^2$

Now, consider the expression $\frac{z|\overline{z}{|}^2}{\overline{z}|z^2|}$

Substituting the above obtained values in the expression we get

$\Rightarrow \frac{z|\overline{z}{|}^2}{\overline{z}|z^2|}=\frac{(x+iy)(x^2+y^2)}{(x-iy)(x^2+y^2)}$

$\Rightarrow \frac{z|\overline{z}{|}^2}{\overline{z}|z^2|}=\frac{(x+iy)}{(x-iy)}$

$\Rightarrow \frac{z|\overline{z}{|}^2}{\overline{z}|z^2|}=\frac{(x+iy)}{(x-iy)}\times \frac{x+iy}{x+iy}$

$\Rightarrow \frac{z|\overline{z}{|}^2}{\overline{z}|z^2|}=\frac{(x+iy{)}^2}{(x^2+y^2)}$

$\Rightarrow \frac{z|\overline{z}{|}^2}{\overline{z}|z^2|}=\frac{x^2+i^2{y}^2+2xyi}{(x^2+y^2)}$

Substituting $\Rightarrow y^2=3x^2$ we get

$\Rightarrow \frac{z|\overline{z}{|}^2}{\overline{z}|z^2|}=\frac{x^2-3x^2+2\sqrt{3}{x}^{2}i}{(x^2+3x^2)}$

$\Rightarrow \frac{z|\overline{z}{|}^2}{\overline{z}|z^2|}=\frac{-2x^2+2\sqrt{3}{x}^{2}i}{4x^2}$

Therefore, $\Rightarrow \frac{z|\overline{z}{|}^2}{\overline{z}|z^2|}=\frac{-1+\sqrt{3}i}{2}$