Question

In Argand's plane, the point corresponding to $\frac {(1-i\sqrt {3})(1+i)} {(\sqrt {3}+i)}$ lies in

• A. quadrant I
• B. quadrant II
• C. quadrant III
• D. quadrant IV

Answer 1

The Correct Option is D

Given,
$\frac{(1-i \sqrt{3})(1+i)}{(\sqrt{3}+i)}=\frac{(1-i \sqrt{3}+i+\sqrt{3})(\sqrt{3}-i)}{(3+1)}$
$\left(\because i^{2}=-1\right)$
$=\frac{1}{4} \cdot\{(1+\sqrt{3})+i(1-\sqrt{3})\} \cdot(\sqrt{3}-i)$
$=\frac{1}{4} \cdot\{\sqrt{3}(1+\sqrt{3})+i(1-\sqrt{3}) \sqrt{3} -(1+\sqrt{3}) i+(1-\sqrt{3})\}$
$\left.=\frac{1}{4} \cdot\{\sqrt{3}+ 3+1-\sqrt{3})+(\sqrt{3}-3-1-\sqrt{3}) i\right\}$
$=\frac{1}{4} \cdot\{4-4 i\}=1-i$
The point $(1-i)$ in Arg and plane is $(1,-1)$ which lies in IVth quadrant