यदि `alpha` और `beta` भिन्न सम्मिश्र संख्याएँ हैं, जहाँ `|beta|=1`, तब `|(beta-alpha)/(1-bar(alpha)beta)|` का मान ज्ञात कीजिए।

Explanation

प्रश्न से, `|beta| = 1 `
अब, ` | ( beta - alpha ) / ( 1 - bar alpha beta ) |^ 2 = ( | beta - alpha |^ 2)/(| 1 - bar alpha beta |^ 2 ) `
` = ((beta - alpha ) bar ( beta - alpha ) ) /( ( 1 - bar alpha beta ) ( bar ( 1 - bar alpha beta)) ) = (( beta - alpha ) (bar beta - bar alpha ) ) /( ( 1 - bar alpha beta ) ( 1 - alpha bar beta ) ) `
` = ( beta bar beta + alpha bar alpha - ( alpha bar beta + beta bar alpha ) ) /( 1- ( bar alpha beta + alpha bar beta ) + ( alpha bar alpha ) ( beta bar beta )) `
` = ( | beta |^2 + | alpha |^ 2 - ( alpha bar beta + beta bar alpha )) /( 1 - (bar alpha beta + alpha bar beta ) + | alpha |^ 2 | beta |^ 2 ) `
` = ( 1 + |alpha|^ 2 - ( alpha bar beta+ beta bar alpha ) ) / ( 1 - ( bar alpha beta + alpha bar beta ) + | alpha |^2) " "[ because |beta | = 1 ] `
= 1