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

If (z-1)/(z+1) is purely imaginary then

If z_1, z_2 are two complex numbers (z_1!=z_2) satisfying |z_1^2-z_2^2|=| 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

Z_1!=Z_2 are two points in an Argand plane. If a|Z_1|=b|Z_2|, then prove that (a Z_1-b Z_2)/(a Z_1+b Z_2) is purely imaginary.

Prove that |z-z_1|^2+|z-z_2|^2=k will represent a real circle with center ((z_1+z_2)/2) on the Argand plane if 2kgeq|z_1-z_2|^2

If z_1 and z_2 be two non-zero complex numbers such that |z_1+z_2|=|z_1|+|z_2| ,then prove that argz_1-argz_2=0 .

P(z_1),Q(z_2),R(z_3)a n dS(z_4) are four complex numbers representing the vertices of a rhombus taken in order on the complex lane, then which one of the following is/ are correct? (z_1-z_4)/(z_2-z_3) is purely real a m p(z_1-z_4)/(z_2-z_3)=a m p(z_2-z_4)/(z_3-z_4) (z_1-z_3)/(z_2-z_4) is purely imaginary It is not necessary that |z_1-z_3|!=|z_2-z_4|

If z_1,z_2, z_3 are complex numbers such that |z_1|=|z_2|=|z_3|=|1/z_1+1/z_2+1/z_3|=1 then |z_1+z_2+z_3| is equal to

If z_1, z_2 are complex number such that (2z_1)/(3z_2) is purely imaginary number, then find |(z_1-z_2)/(z_1+z_2)| .

If z_(1)and z_(2) be any two non-zero complex numbers such that |z_(1)+z_(2)|=|z_(1)|+|z_(2)| then prove that, arg z_(1)=arg z_(2) .

If the vertices of an equilateral traingle be represented by the complex numbers z_1 , z_2 , z_3 , then prove that z_1^2+z_2^2+z_3^2=3z_0^2 and z_0 be the circumcenter of the triangle.