Question

If $z_1,z_2.z_3$ are three nonzero complex numbers such that $z_3=(1-\lambda )z_1+\lambda z_2$ where $\lambda \in R-0$, then prove that the points corresponding to $z_1,z_2$ and $z_3$ are collinear.

Answer 1

$z_3=(1-\lambda )z_1+z_3=\frac{(1-\lambda )z_1+\lambda z_2}{1-\lambda +\lambda }$
Hence,
$z_3$ divides the line joining $A\left(z_1\right)$ and $B\left(z_2\right)$ in the ratio $\lambda :(1-\lambda )$. Thus, the given points are collinear.
Ans: 1