`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_1+z_2|^2=|z_1|^2, ifz_1//z_2 is purely imaginary.

If |z_1|=|z_2|=1, then prove that |z_1+z_2| = |1/z_1+1/z_2∣

z_1a n dz_2 are two distinct points in an Argand plane. If a|z_1|=b|z_2|(w h e r ea ,b in R), then the point (a z_1//b z_2)+(b z_2//a z_1) is a point on the line segment [-2, 2] of the real axis line segment [-2, 2] of the imaginary axis unit circle |z|=1 the line with a rgz=tan^(-1)2

Given that |z_1+z_2|^2=|z_1|^2+|z_2|^2 , prove that z_1/z_2 is purely imaginary.

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

If |z/| barz |- barz |=1+|z|, then prove that z is a purely imaginary number.

If z_1 and z_2 are two complex numbers such that |z_1|lt1lt|z_2| then prove that |(1-z_1barz_2)/(z_1-z_2)|lt1

Let z_1, z_2 be two complex numbers with |z_1| = |z_2| . Prove that ((z_1 + z_2)^2)/(z_1 z_2) is real.

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

Let Z_1 and Z_2 are two non-zero complex number such that |Z_1+Z_2|=|Z_1|=|Z_2| , then Z_1/Z_2 may be :