`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.

If |z_1+z_2| = |z_1-z_2| , prove that amp z_1 - amp z_2 = pi/2 .

If |z_1|=1,|z_2|=1 then prove that |z_1+z_2|^2+|z_1-z_2|^2 =4.

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 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 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

Find the locus of point z in the Argand plane if (z-1)/(z+1) is purely imaginary.

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