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)=a will represent a real circle Iwith center (|Z_(1)+Z_(2)|/2+) on the Argand plane if 2a>=|Z_(1)-Z_(1)|^(2)

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

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

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

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 |z_1+z_2|^2=|z_1|^2+|z_2|^2, ifz_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.

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

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