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 represnts `z_1` and `z_2` in the complex plane if `leAOB=alpha ne 0` and OA=OB where 0 is the origin prove that `p^2=4qcos^2(alpha/2)`

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 represnts z_1 and z_2 in the complex plane if angleAOB=alpha ne 0 and OA=OB where 0 is the origin prove that p^2=4qcos^2(alpha/2)

If one root of the equation z^2-a z+a-1= 0 is (1+i), where a is a complex number then find the root.

The origin and the roots of the equation z^2+pz+q=0 form an equilateral triangle if

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

If z ne 1 and (z^2)/(z-1) is real then the point represented by the complex number z lies

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

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

Let p and q are complex numbers such that |p|+|q| lt 1 . If z_(1) and z_(2) are the roots of the z^(2)+pz+q=0 , then which one of the following is correct ?

Let z_(1) and z_(2) be the roots of z ^(2)+az+b=0 . If the origin, z_(1) and z_(2) form an equilateral triangle. Then-

Let the complex numbers z_1,z_2 and z_3 be the vertices of an equilateral triangle let z_0 be the circumcentre of the triangle then prove that z_1^2+z_2^2+z_3^2=3z_0^2