Question

If the points A(z), B(-z), and C(1-z) are the vertices of an equilateral triangle ABC, then

• A. Sum of possible z is 1/2
• B. Sum of possible z is 1
• C. Product of possible z is 1/4
• D. Product of possible z is 1/2

Answer 1

Answer: (A), (C)

The correct options are
A Sum of possible z is 1/2
C Product of possible z is 1/4

Since it is an equilateral triangle.
$z_1^2+z_2^2+z_3^2=z_1{z}_2+z_2{z}_3+z_3{z}_1$
Hence
$z^2+(-z{)}^2+(1-z{)}^2=z(-z)+z(1-z)-z(1-z)$
$3z^2-2z+1=-z^2+z-z^2-z+z^2$
$4z^2-2z+1=0$
Sum of roots
$=-\frac{2}{4}$
$=\frac{1}{2}$.
Product of roots,
$=\frac{1}{4}$