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Q.
If $A\left(z_{1}\right), B\left(z_{2}\right)$ and $C\left(z_{3}\right)$ are collinear points then
Complex Numbers and Quadratic Equations
Solution:
If $A\left(z_{1}\right), B\left(z_{2}\right)$ and $C\left(z_{3}\right)$ are collinear, then $\arg \left(\frac{z_{3}-z_{1}}{z_{2}-z_{1}}\right)=0, \pi$
$\Rightarrow \frac{z_{3}-z_{1}}{z_{2}-z_{1}}$ is purely real
$\Rightarrow \frac{z_{3}-z_{1}}{z_{2}-z_{1}}=\frac{\bar{z}_{3}-\bar{z}_{1}}{\bar{z}_{2}-\bar{z}_{1}}$
$\Rightarrow \left(z_{3}-z_{1}\right)\left(\bar{z}_{2}-\bar{z}_{1}\right)$
$=\left(z_{2}-z_{1}\right)\left(\bar{z}_{3}-\bar{z}_{1}\right)$