Solve the following

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 _____________________

Solution:

Let $z_{1}=\sqrt{3}(1+i)$ and $z_{2}=\sqrt{3}+i$

then $\frac{z_{1}}{z_{2}}=\frac{\sqrt{3}(1+i)}{\sqrt{3}+i} \times \frac{\sqrt{3}-i}{\sqrt{3}-i}$

$=\frac{\sqrt{3}(1+i)(\sqrt{3}-i)}{3-i^{2}}$

$=\frac{\sqrt{3}\left(\sqrt{3}-i+\sqrt{3} i-i^{2}\right)}{3+1}$

$=\frac{\sqrt{3}}{4}(\sqrt{3}+1+i(\sqrt{3}-1))$

Since $\sqrt{3}+1$ and $\sqrt{3}-1>0$

$\Rightarrow \frac{z_{1}}{z_{2}}$ lies in I quadrant

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