Question

The complex numbers $z_1, z_2$ and $z_3$ satisfying $\frac{z_1-z_3}{z_2-z_3}=\frac{1-i \sqrt 3}{2}$ are the vertices of a triangle which is

• A. of area zero
• B. right angled isosceles
• C. equilateral
• D. obtuse angled isosceles

Answer 1

Answer: (C)

equilateral

Answer 2

$\frac{z_{1}-z_{3}}{z_{2}-z_{3}}=\frac{1-i \sqrt{3}}{2}$
$=\cos \left(\frac{-\pi}{3}\right)+\sin \left(\frac{-\pi}{3}\right)=e^{-i} \pi^{3}$
$\therefore\left|\frac{z_{1}-z_{3}}{z_{2}-z_{3}}\right|=\left|e^{-i \pi / 3}\right|=1$
and angle between $z_{1}-z_{3}$ and $z_{2}-z_{3}$ is $\frac{\pi}{3}$.
$\therefore$ triangle is equilateral.