`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.

Let z_1 nez_2 are two points in the Argand plane if a abs(z_1)=b abs(z_2) then point (az_1-bz_2)/(az_1+bz_2) lies

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

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

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=x+iy and |z|=1 , show that (z-1)/(z+1)(z!=-1) is a purely imaginary quantity.

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 .

If Z_1 = -3 + 4i , Z_2 = 12 - 5i , prove that |z_1+z_2|<|z_1-z_2|

if z_1 and z_2 are two complex numbers such that absz_1lt1ltabsz_2 then prove that abs(1-z_1barz_2)/abs(z_1-z_2)lt1