`|Z-(Z_1+Z_1)/2|^2+|(Z_1-Z_2)/2|^2=`

If z_1, z_2 in C , then say which are true and false - . |z_1+z_2|^2=|z_1""|^2+|z_2|^2-2R e(z_1 z_2) |z_1-z_2|^2=|z_1""|^2-|z_2|^2-2R e(z_1 z_2) |z_1+z_2|^2+|z_1-z_2|^2=2(|z_1|^2+|z_2|^2) |a z_1-b z_2|^2+|b z_1+a z_2|^2=(a^2+b^2)(|z_1|^2+|z_2|^2) , where a ,b in Rdot

For any two complex numbers z_1 and z_2 prove that: |\z_1+z_2|^2 +|\z_1-z_2|^2=2[|\z_1|^2+|\z_2|^2]

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

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 [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 [with center (|Z_1+Z_2|^//2+) ] on the Argand plane if 2ageq|Z_1-Z_1|^2

If u= sqrt(z_1 z_2) , prove that |z_1|+|z_2|=|(z_1+z_2)/2+u|+|(z_1+z_2)/2-u| .

If |z_1|le1,|z_2|le1"show that" |1-z_1z_2|^2-|z_1-z_2|^2=(1-|z_1|^2)(1-|z_2|^2 "Hence or otherwise show that." |(z_1-z_2)/(1-z_1z_2)|lt 1"if"|z_1| lt 1,|z_2| lt 1

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