Question

lf three points having affixes $z_1,z_2,z_3$ are connected by the relation $a{z}_1+b{z}_2+c{z}_3=0$ where $a,b,cϵR$ such that $a+b+c=0,$ then?

• A. $z_1,z_2,z_3$ are collinear
• B. $z_1,z_2,z_3$ are vertices ofan equilateral triangle
• C. $z_1,z_2,z_3$ form right angle $\Delta$
• D. None of these

Answer 1

Answer: (A)

$z_1,z_2,z_3$ are collinear

We have
$a{z}_1+b{z}_2+c{z}_3=0$
And $a+b+c=0$
Therefore
$a{z}_1+b{z}_1+c{z}_1=0$
Subtracting, we get
$b(z_2-z_1)+c(z_3-z_1)=0$
$z_{2}b+z_{3}c=z_1(b+c)$
Hence
$z_1=\frac{z_{2}b+z_{3}c}{b+c}$
Hence
$z_1,z_2,z_3$ are collinear.