If `|z|=2a n d(z_1-z_3)/(z_2-z_3)=(z-2)/(z+3)` , then prove that `z_1, z_2, z_3` are vertices of a right angled triangle.

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 a r g["z"_1("z"_3-"z"_2)]="a r g"["z"_3("z"_2-"z"_1)] , then find prove that O ,z_1, z_2, z_3 are concyclic, where O is the origin.

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.

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

Complex numbers z_1 , z_2 , z_3 are the vertices A, B, C respectively of an isosceles right angled trianglewith right angle at C and (z_1- z_2)^2 = k(z_1 - z_3) (z_3 -z_2) , then find k.

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_1^2+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.

The points z_(1)z_(2)z_(3)z_(4) in the complex plane are the vertices of a parallelgram taken in order if and only if :

If the complex numbers z_(1), z_(2)" and "z_(3) denote the vertices of an isoceles triangle, right angled at z_(1), " then "(z_(1)-z_(2))^(2)+(z_(1)-z_(3))^(2) is equal to