Question

किन्ही दो सम्मिश्र संख्याएँ ` z _ 1 ` तथा ` z _ 2 ` के लिए साबित कीजिये कि
` |z _ 1 + z _ 2 |^ 2 + | z _ 1 - z _ 2 |^ 2 = 2 [|z _ 1 |^ 2 + | z _ 2 |^ 2 ] `

Answer 1

LHS = ` | z _ 1 + z _ 2 |^ 2 + | z _ 1 - z _ 2 |^ 2 `
` = ( z _ 1 + z _ 2 ) ( bar ( z _ 1 + z _ 2 ) ) + ( z _ 1 - z _ 2 ) ( bar ( z _ 1 - z _ 2 )) " " [ because |z|^ 2 = z bar z] `
= ` ( z _ 1 + z _ 2) ( bar z _ 1 + bar z _ 2 ) + ( z _ 1 - z _ 2 ) ( bar z _ 1 - bar z _ 2 ) `
` = ( z _ 1 bar z _1 + z _ 2 bar z _ 1 ) + z _ 1 bar z _ 2 + z _ 2 bar z _ 2 ) + (z _ 1 bar z _1 - z _ 2 bar z _ 1 - z_ 1 bar z _ 2 + z _ 2 bar z _ 2 ) `
` = 2 ( z _ 1 bar z _ 1 + z _ 2 bar z _ 2 ) = 2 (| z _ 1 | ^ 2 + | z _ 2|^2 ) `