If `z_1` and `z_2` are two complex numbers then prove that `|z_1-z_2|^2lt=(1+k)z_1^2+(1+1/k)z_2^2`

If z_1 and z_2 are two complex numbers, then prove that |z_1 - z_2|^2 le (1+ k)|z_1|^2 +(1+K^(-1) ) |z_2|^2 AA k in R^(+) .

If z_1 and z_2 are two complex numbers and c >0 , then prove that |z_1+z_2|^2lt=(1+c)|z_1|^2+(1+c^(-1))|z_2|^2dot

If z_1 , and z_2 be two complex numbers prove that z_1barz_1=|z_1|^2

If z_1 and z_2 (ne 0) are two complex numbers, prove that : |z_1 z_2|= |z_1||z_2| .

If z_1 , and z_2 be two complex numbers prove that bar(z_1+-z_2)=barz_1+-barz_2

If z_(1),z_(2) are two complex numbers , prove that , |z_(1)+z_(2)|le|z_(1)|+|z_(2)|

If z_1 , and z_2 be two complex numbers prove that |z_1+z_2|^2+|z_1-z_2|^2=2[|z_1|^2+|z_2|^2]

If z_1 and z_2 (ne 0) are two complex numbers, prove that : |z_1/z_2|= (|z_1|)/ (|z_2|), z_2 ne 0 .

If z_(1) and z_(2) are two complex numbers such that |(z_(1)-z_(2))/(z_(1)+z_(2))|=1, then