Question

Let $z_1,z_2,z_3$ be three complex numbers and a, b, c be real numbers not all zero, such that $a+b+c=0$ and $a{z}_1+b{z}_2+c{z}_3=0$, then prove that $z_1,z_2,z_3$ are collinear.

Answer 1

Given,
$a+b+c=0$ (1)
and $a{z}_1+b{z}_2+z_3=0$ (2)
Since a, b, c are not all zero, from (2), we have
$a{z}_1+b{z}_2-(a+b)z_3=0$ [From (1), $c=-(a+b)$]
or $a{z}_1+b{z}_2=(a+b)z_3$
or $z_3=\frac{a{z}_1+b{z}_2}{a+b}$ (3)
From (3), it follows that $z_3$ divides the line segment joing $z_1$ and $z_3$ internally in the ratio $b:a$
If $a$ and $b$ are of the same sign, then division is in fact internal, and if $a$ and $b$ are of opposite sign, then division is external in the ratio $\left|b\right|:\left|a\right|$.
Therefore, $z_1,z_2$ and $z_3$ are collinear.
Ans: 1