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

• A. $z_1,z_2,z_3$ are vertices of a triangle.
• B. $z_1,z_2,z_3$ lies on circumference of a circle
• C. $z_1,z_2,z_3$ are collinear.
• D. None of these.

Answer 1

The correct option is C $z_1,z_2,z_3$ are collinear.

Given,
$a+b+c=\mathrm{0....}\left(i\right)$
and
$a{z}_1+b{z}_2+c{z}_3=\mathrm{0....}\left(ii\right)$
Since $a,b,c$ are not all zero, from $\left(i\right)$ and $\left(ii\right)$, we have
$a{z}_1+b{z}_2-(a+b)z_3=0$
$\Rightarrow a{z}_1+b{z}_2=(a+b)z_3$
$\Rightarrow z_3=\frac{a{z}_1+b{z}_2}{a+b}....\left(iii\right)$
From $\left(iii\right)$, it follows that $z_3$ divides the line segment joining $z_1$ and $z_2$ internally in the ratio $b:a$. If $a$ and $b$ are of opposite sign, then division is external in the ratio $|b|:|a|$.