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

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

Prove that |z_1+z_2|^2+|z_1-z_2|^2 =2|z_1|^2+2|z_2|^2 .

Prove that |1-barz_1z_2|^2-|z_1-z_2|^2=(1-|z_1|^2)(1-|z_2|^2) .

If z_1, z_2 are two complex numbers (z_1!=z_2) satisfying |z1^2-z2^2|=| z 1^2+ z 2 ^2-2( z )_1( z )_2| , then a. (z_1)/(z_2) is purely imaginary b. (z_1)/(z_2) is purely real c. |a r g z_1-a rgz_2|=pi d. |a r g z_1-a rgz_2|=pi/2

If the triangle fromed by complex numbers z_(1), z_(2) and z_(3) is equilateral then prove that (z_(2) + z_(3) -2z_(1))/(z_(3) - z_(2)) is purely imaginary number

If z_(1) and z_(2) are complex numbers such that |z_(1)|=|z_(2)|=1. If z_(1),z_(2) have purely imaginary product and purely real quotient then the no.of ordered pair of (z_(1),z_(2)) is :

Prove that |(z_1, z_2)/(1-barz_1z_2)|lt1 if |z_1|lt1,|z_2|lt1

If z^(2) + |z|^(2) = 0 , show that z is purely imaginary.

Let A(z_1),B(z_2) and C(z_3) be the vertices of an equilateral triangle in the Argand plane such that |z_1|=|z_2|=|z_3|. Then (A) (z_2+z_3)/(2z_1-z_2-z_3) is purely real (B) (z_2-z_3)/(2z_1-z_2-z_3) is purely imaginary (C) |arg(z_1/z_2)|=2 arg((z_3-z_2)/(z_1-z_2))| (D) none of these

If z_1 and z_2 are two complex numbers, then prove that |z_1 - z_2|^2 le (1+ k)|z_1|^2 +(1+K^(-1) ) |z_2|^2 AA k in R^(+) .