Question

If $a,b,c$ are real and $a+b+c=0$ (for at least one of $a,b,c$ non zero ) and $a{z}_1+b{z}_2+c{z}_3=0$ then $z_1,z_2,z_3$ are

• A. Vertices of equilateral triangle
• B. Vertices of an isosceles triangle
• C. Vertices of right angled triangle
• D. Collinear points

Answer 1

The correct option is D Collinear points

The necessary and sufficient condition for the points $z_1,z_2$ & $z_3$ to be collinear is $\frac{z_1-z_3}{z_2-z_3}$ ie purely real.
Given that $a{z}_1+b{z}_2+c{z}_3=0$
$a{z}_1+b{z}_2+(-a-b)z_3=0$ [ $a+b+c=0$ ]

$a(z_1-z_3)+b(z_2-z_3)=0$

$\frac{z_1-z_3}{z_2-z_3}=\frac{-b}{a}$ [$\therefore a,b,c$ are real]

$\frac{-b}{a}$ is purely real

Therefore $z_1,z_2$ & $z_3$ are collinear points.