• # question_answer For any two complex numbers ${{z}_{1}}$and${{z}_{2}}$ and any real numbers a and b; $|(a{{z}_{1}}-b{{z}_{2}}){{|}^{2}}+|(b{{z}_{1}}+a{{z}_{2}}){{|}^{2}}=$ [IIT 1988] A) $({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}|+|{{z}_{2}}|)$ B) $({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}})$ C) $({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}{{|}^{2}}-|{{z}_{2}}{{|}^{2}})$ D) None of these

$|(a{{z}_{1}}-b{{z}_{2}}){{|}^{2}}+|(b{{z}_{1}}+a{{z}_{2}}){{|}^{2}}$ $={{a}^{2}}|{{z}_{1}}{{|}^{2}}+{{b}^{2}}|{{z}_{2}}{{|}^{2}}-2\operatorname{Re}(ab)|{{z}_{1}}{{\overline{z}}_{2}}|+{{b}^{2}}|{{z}_{1}}{{|}^{2}}+$ ${{a}^{2}}|{{z}_{2}}{{|}^{2}}+2\operatorname{Re}(ab)|{{\overline{z}}_{1}}{{z}_{2}}|$ $=({{a}^{2}}+{{b}^{2}})(|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}})$