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 |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_0|=|z_2-z_0|=a and amp((z_2-z_0)/(z_0-z_1))=pi/2 , then find z_0

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.

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

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.

Prove that |z_1+z_2|^2=|z_1|^2+|z_2|^2, ifz_1//z_2 is purely imaginary.

If |z_1-1|lt=1,|z_2-2|lt=2,|z_(3)-3|lt=3, then find the greatest value of |z_1+z_2+z_3|dot

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.

If z_1, z_2 in C ,z_1^ 2 in R ,z_1(z_1^ 2-3z_2 ^2)=2 a n d z_2(3z_1 ^2-z_2^ 2)=11 , then the value of z_1^ 2+z_2^ 2 is a. 10 b. 12 c. 5 d. 8