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`

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_(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 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 and B represent z_(1) and z_(2) in the Argand plane and z_(1),z_(2) be the roots of the equation Z^(2)+pZ+q=0 ,where p, q are complex numbers. If O is the origin, OA=OB and /_AOB=alpha then p^(2)=

If z_(1),z_(2) are the non zero complex root of z^(2)-ax+b=0 such that |z_(1)|=|z_(2)|, where a,b arecomplex numbers.If A(z_(1)),B(z_(2)) and /_AOB=theta,'O' being the origin,then prove a^(2)=4b cos^(2)((theta)/(2))

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 z_1 and z_2 be theroots of the equation z^2+az+b=0 z being complex. Further, assume that the origin z_1 and z_2 form an equilateral triangle then (A) a^2=4b (B) a^2=b (C) a^2=2b (D) a^2=3b

Let z_1a n dz_2 be two complex numbers such that ( z )_1+i( z )_2=0a n d arg(z_1z_2)=pidot Then, find a r g(z_1)dot

Let A, B, C represent the complex numbers z_1, z_2, z_3 respectively on the complex plane. If the circumcentre of the triangle ABC lies at the origin, then the orthocentre is represented by the complex number