Prove that `|z_(1) + z_(2)|^(2)= |z_(1)|^(2) + |z_(2)|^(2)`, if `z_(1)//z_(2)` is purely imaginary

If z_(1) and z_(2) are two non-zero complex numbers such that |z_(1) + z_(2)| = |z_(1)| + |z_(2)| , then arg ((z_(1))/(z_(2))) is equal to a)0 b) -pi c) -(pi)/(2) d) (pi)/(2)

Let z_(1) = 3 + 4i and z_(2) = - 1 + 2i . Then, |z_(1) + z_(2)|^(2) - 2(|z_(1)|^(2) + |z_(2)|^(2)) is equal to

If z_(1)=2+3i and z_(2)=3+2i , then |z_(1)+z_(2)| is equal to

If z_(1) and z_(2) are complex numbers such that z_(1) ne z_(2) and |z_(1)|= |z_(2)| . If z_(1) has positive real part and z_(2) has negative imaginary part, then ((z_(1) +z_(2)))/((z_(1)-z_(2))) may be

If z_(1), z_(2),……..,z_(n) are complex numbers such that |z_(1)| = |z_(2)| = …….. = |z_(n)| = 1 , then |z_(1) + z_(2) +……..+ z_(n)| is equal to a) |z_(1)z_(2)z_(3)…..z_(n)| b) |z_(1)|+|z_(2)|+…….+|z_(n)| c) |(1)/(z_(1)) + (1)/(z_(2)) + ……….+ (1)/(z_(n))| d)n

Given that for any complex number z, absz^2 = zoverlinez . Prove that abs(z_1+z_2)^2 + abs(z_1-z_2)^2 = 2[abs(z_1)^2 + abs(z_2)^2] where z_1 and z_2 are any two complex numbers.

Let x_1 and y_1 be real numbers. If z_1 and z_2 are complex numbers such that |z_1| = |z_2|=4 , then |x_1 z_1 - y_1 z_2|^(2) + |y_1 z_1 + x_1 z_2 |^(2) is equal to