First slide

Geometry of complex numbers

Question

If $a, b, c$ are three real numbers such that $a + b + c = 0$ (at least one $a, b, c$ is not equal to zero) and $a z_{1} + b z_{2} + c z_{3} = 0,$ then $z_{1}, z_{2}, z_{3}$ :

Moderate

A

are collinear
B

form an equilateral triangle
C

form a scalene triangle
D

form a isosceles triangle

Solution

$a z_{1} + b z_{2} + c z_{3} = 0,$ $a + b + c = 0.$ at least one of $a, b, c \neq 0$ and $a, b, c \in R$ . Let $a \neq 0$
Then $z_{1} = \frac{b z_{2} + c z_{3}}{b + c}$ $[∵ - a = b + c]$

Thus, $z_{1}$ divides the line segment joining $z_{2}$ and $z_{3}$ in the ratio $c : b$ and therefore $z_{1}, z_{2}, z_{3}$ are collinear

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