Question

The three points $z_1$, $z_2$, $z_3$, are connected by the relation $a{z}_1+b{z}_2+c{z}_3=0$, $z_1$, $z_2$, $z_3$ are complex numbers and a + b + c = 0. Then the points are :

• A. Collinear
• B. Non collinear
• C. Linearly dependent
• D. Linearly Independent

Answer 1

Answer: (A)

Collinear

$a{z}_1+b{z}_2+c{z}_3=0$
$a+b+c=0$
$\therefore c=-a-b$
$\therefore a{z}_1+b{z}_2+(-a-b)z_3=0$
$\therefore (a+b)z_3=a{z}_1+b{z}_2$
$\therefore z_3=\frac{a{z}_1+b{z}_2}{a+b}$
Thus $z_3$ is a point which which divides the line segment joining $z_1$ & $z_2$ in the ratio b:a
$\Rightarrow z_3$ lies on the line segment joining $z_1$ & $z_2$
$\Rightarrow z_1,z_2,z_3$ are collinear!