Question

If arg $\frac{z_1-z_2}{z_1-z_3}=\frac{\pi }{2}$, where $\left|z_1\right|\ne \left|z_2\right|\ne \left|z_3\right|$, then the points representing $z_1$, $z_2$ and $z_3$ are

• A. collinear
• B. vertices of an equilateral triangle
• C. vertices of a right-angled triangle
• D. vertices of an isosceles right-angled triangle

Answer 1

Answer: (C)

vertices of a right-angled triangle

Let
$z_1=A$
$z_2=B$
$z_3=C$
Hence,
$arg\left(\frac{z_1-z_2}{z_1-z_3}\right)$
$=\angle A$
$=\frac{\pi }{2}$
Hence, $z_1,z_2,z_3$ are the vertices of a right angled triangle.
Since, none of the sides are equal, the triangle cannot be isosceles.
Hence, option C is correct.