Let and be roots of the equation

MATHEMATICS · Subjective · CBSE · Mathematics · Complex Numbers and Quadratic Equations · Complex Numbers and Quadratic Equations

Problem

Let

z_1

and

z_2

be roots of the equation

z^2 + pz + q = 0

, where the coefficients

p

and

q

may be complex numbers. Let

A

and

B

represent

z_1

and

z_2

in the complex plane. If

\triangle AOB = \eta \neq 0

and

OA = OB

, where

O

is the origin, prove that

p^2 = 4q \cos^2(\alpha/2)