Let `z_1 and z_2` be the root of the equation `z^2+pz+q=0` where the coefficient p and q may be complex numbers. Let A and B represent `z_1 and z_2` in the complex plane. If `/_AOB=alpha!=0 and 0 and OA=OB, where O` is the origin prove that `p^2=4qcos^2 (alpha/2)`

Let z_1a n dz_2 b be toots of the equation z^2+p z+q=0, where he coefficients pa n dq may be complex numbers. Let Aa n dB represent z_1a n dz_2 in the complex plane, respectively. If /_A O B=theta!=0a n dO A=O B ,w h e r eO is the origin, prove that p^2=4q"cos"^2(theta//2)dot

z_(1) and z_(2) are the roots of the equaiton z^(2) -az + b=0 where |z_(1)|=|z_(2)|=1 and a,b are nonzero complex numbers, then

Let z_(1) and z_(2) be the roots of the equation z^(2)+pz+q=0 . Suppose z_(1) and z_(2) are represented by points A and B in the Argand plane such that angleAOB=alpha , where O is the origin. Statement-1: If OA=OB, then p^(2)=4q cos^(2)alpha/2 Statement-2: If affix of a point P in the Argand plane is z, then ze^(ia) is represented by a point Q such that anglePOQ =alpha and OP=OQ .

Let z_1 and z_2 be the roots of the equation 3z^2 + 3z + b=0 . If the origin, together with the points represented by z_1 and z_2 form an equilateral triangle then find the value of b.

Number of solutions of the equation z^(3)+(3(bar(z))^(2))/(|z|)=0 where z is a complex number is

Solve the equation z^(2)+|z|=0, where z is a complex number.

Let z_1 and z_2 be the roots of z^2+pz+q =0. Then the points represented by z_1,z_2 the origin form an equilateral triangle if p^2 =3q.

Let a , b , c be distinct complex numbers with |a|=|b|=|c|=1 and z_(1) , z_(2) be the roots of the equation az^(2)+bz+c=0 with |z_(1)|=1 . Let P and Q represent the complex numbers z_(1) and z_(2) in the Argand plane with /_POQ=theta , o^(@) lt 180^(@) (where O being the origin).Then