Let `z_1 \and\ z_2` be the roots of the equation `z^2+p z+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, respectively. If `/_A O B=theta!=0 \and \O A=O B ,\where\ O` 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

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.

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

Let z_1 and z_2 be theroots of the equation z^2+az+b=0 z being compex. Further, assume that the origin z_1 and z_2 form an equilatrasl triangle then

Roots of the equation x^2 + bx + 45 = 0 , b in R lie on the curve |z + 1| = 2sqrt(10), where z is a complex number 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 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

Let z_1, z_2 be two complex numbers represented by points on the circle |z_1|= and and |z_2|=2 are then

If A and B represent the complex numbers z_(1) and z_(2) such that |z_(1)+z_(2)|=|z_(1)-z_(2)| , then the circumcenter of triangleOAB , where O is the origin, is

Let OP.OQ=1 and let O,P and Q be three collinear points. If O and Q represent the complex numbers of origin and z respectively, then P represents