If `z_1^2+z_2^2+2z_1.z_2.costheta= 0 ` prove that the points represented by `z_1, z_2`, and the origin form an isosceles triangle.

Let z_(1)" and "z_(2) be the roots of the equation z^(2)+pz+q=0 , where p,q are real. The points represented by z_(1),z_(2) and the origin form an equilateral triangle if

If z_1, z_2, z_3 are complex numbers such that (2//z_1)=(1//z_2)+(1//z_3), then show that the points represented by z_1, z_2()_, z_3 lie on a circle passing through the origin.

If |z_(1)+z_(2)|=|z_1|+|z_2| then

If ((3-z_1)/(2-z_1))((2-z_2)/(3-z_2))=k(k >0) , then prove that points A(z_1),B(z_2),C(3),a n dD(2) (taken in clockwise sense) are concyclic.

If |z_1/z_2|=1 and arg (z_1z_2)=0 , then

If |z|=2a n d(z_1-z_3)/(z_2-z_3)=(z-2)/(z+2) , then prove that z_1, z_2, z_3 are vertices of a right angled triangle.

If z_1, z_2, z_3 are three nonzero complex numbers such that z_3=(1-lambda)z_1+lambdaz_2w h e r elambda in R-{0}, then prove that points corresponding to z_1, z_2a n dz_3 are collinear .

Let the altitudes from the vertices A, B and Cof the triangle e ABCmeet its circumcircle at D, E and F respectively and z_1, z_2 and z_3 represent the points D, E and F respectively. If (z_3-z_1)/(z_2-z_1) is purely real then the triangle ABC is

If z+1//z=2costheta, prove that |(z^(2n)-1)//(z^(2n)+1)|=|tanntheta|

Let z_1, z_2, z_3 be the three nonzero complex numbers such that z_2!=1,a=|z_1|,b=|z_2|a n d c=|z_3|dot Let |(a, b, c), (b, c, a), (c,a,b) |=0 a r g(z_3)/(z_2) equal to (a) arg((z_3-z_1)/(z_2-z_1))^2 (b) orthocentre of triangle formed by z_1, z_2, z_3, i sz_1+z_2+z_3 (c)if triangle formed by z_1, z_2, z_3 is equilateral, then its area is (3sqrt(3))/2|z_1|^2 (d) if triangle formed by z_1, z_2, z_3 is equilateral, then z_1+z_2+z_3=0