Question

If $z_1=\sqrt{3}+i\sqrt{3}$ and $z_2=\sqrt{3}+i,$ then the point representing $\frac{z_1}{z_2}$ lies in ____________.

Answer 1

$\mathrm{Let}{z}_1=\sqrt{3}\left(1+i\right)\mathrm{and}{z}_2=\sqrt{3}+i$

$\begin{array}{rcl}\mathrm{then}\frac{z_1}{z_2}& =& \frac{\sqrt{3}\left(1+i\right)}{\sqrt{3}+i}\times \frac{\sqrt{3}-i}{\sqrt{3}-i}\\ & =& \frac{\sqrt{3}\left(1+i\right)\left(\sqrt{3}-i\right)}{3-i^2}\\ & =& \frac{\sqrt{3}\left(\sqrt{3}-i+\sqrt{3}i-i^2\right)}{3+1}\\ & =& \frac{\sqrt{3}}{4}\left(\sqrt{3}+1+i\left(\sqrt{3}-1\right)\right)\end{array}$

$$\begin{gathered} \mathrm{Since}\sqrt{3}+1\mathrm{and}\sqrt{3}-1>0 \\ \Rightarrow \frac{z_1}{z_2}\mathrm{lies}\mathrm{in}\mathrm{I}\mathrm{quadrant} \end{gathered}$$