Prove that `|Z-Z_1|^2+|Z-Z_2|^2=a` will represent a real circle [with center `(|Z_1+Z_2|^//2+)` ] on the Argand plane if `2ageq|Z_1-Z_1|^2`

Prove that |z-z_1|^2+|z-z_2|^2 = k will represent a circle if |z_1-z_2|^2 le 2k.

Show that complex numbers z_1=-1+5i and z_2=-3+2i on the argand plane.

The centre of circle represented by |z+1|=2|z-1| in the complex plane is

If the equation |z-z_1|^2 + |z-z_2|^2 =k , represents the equation of a circle, where z_1 = 3i, z_2 = 4+3i are the extremities of a diameter, then the value of k is

Prove that |z_1+z_2|^2+|z_1-z_2|^2 =2|z_1|^2+2|z_2|^2 .

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

Prove that |1-barz_1z_2|^2-|z_1-z_2|^2=(1-|z_1|^2)(1-|z_2|^2) .

Let z be not a real number such that (1+z+z^(2))/(1-z+z^(2))in R, then prove tha |z|=1

Let z, and z_(1)=(a)/(1-1),z_(2)=(b)/(2+1) and z_(3)=a-ib for a,b in R. If z_(1),z_(2)=1, then the centroid of the triangle formed by the points z_(1),z_(2) and z_(3), in the argand plane is given by