Question

The complex numbers $z_1,z_2,z_3$ are such that $\frac{z_1-z_3}{z_2-z_3}=\frac{1-i\sqrt{3}}{2}$ form?

• A. equilateral triangle
• B. isosceles triangle
• C. right angled triangle
• D. collinear points

Answer 1

The correct option is A equilateral triangle

$z_1-z_3=e^{i\pi /3}(z_2-z_3)$
$\Rightarrow arg(z_1-z_3)-arg(z_2-z_3)=\pi /3={60}^0$--------(1)
also, $|z_1-z_3|=\left|e^{i\pi /3}\right||z_2-z_3|$
$\Rightarrow |z_1-z_3|=|z_2-z_3|$-------(1)
So,
The figure has two sides equal and enclosed angle of ${60}^0$
Hence, it is an equilateral triangle.