If $\mathrm{A}\left(\mathrm{z}_1\right)$ and $\mathrm{B}\left(\mathrm{z}_2\right)$ be two complex nu

Question

If $\mathrm{A}\left(\mathrm{z}_1\right)$ and $\mathrm{B}\left(\mathrm{z}_2\right)$ be two complex numbers satisfying the equation $\mathrm{z}+(\sqrt{3}+\mathrm{i}) \mathrm{t}-\mathrm{i}=0$ and $\sqrt{3} \mathrm{z}+(1+\mathrm{i} \sqrt{3}) \lambda-\sqrt{3} \mathrm{i}=0$ respectively $\forall \mathrm{t}, \lambda \in \mathrm{R}$. Let for some $\mathrm{z}_1$ and $\mathrm{z}_2, \arg \left(\mathrm{z}_1\right)=\frac{\pi}{4}$ and $\arg \left(z_2\right)=-\frac{3 \pi}{4} \quad$, then the area of the triangle $A B C$ (where $C$ is point $0+i)$ is

Professor PC Thomas Classes · Answer

$$
\begin{aligned}
& \mathrm{z}_1: \mathrm{z}+(\sqrt{3}+\mathrm{i}) \mathrm{t}-\mathrm{i}=0 	ext { and } \arg \left(\mathrm{z}_1ight)=\frac{\pi}{4} \
& \Rightarrow \operatorname{Re}\left(\mathrm{z}_1ight)=\frac{\sqrt{3}}{2}(\sqrt{3}+1) \
& 	ext { and } \mathrm{z}_2=\mathrm{z}+\left(\frac{\sqrt{3}+\mathrm{i}}{\sqrt{3}}ight) \lambda-\mathrm{i}=0 \Rightarrow \operatorname{Re}\left(\mathrm{z}_2ight)=-\frac{\sqrt{3}+1}{2}
\end{aligned}
$$
Hence area of $	riangle \mathrm{ABC}=\frac{1}{2} 	imes 1 	imes\left(\frac{\sqrt{ } 3+1}{2}ight)(\sqrt{3}+1)=\frac{\sqrt{ } 3+2}{2}$.