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Misc 13 If (𝑎+𝑖𝑏)(𝑐+𝑖𝑑)(𝑒+𝑖𝑓)(𝑔+𝑖ℎ)=𝐴+𝑖𝐵, then show that (𝑎2 + 𝑏2) (𝑐2 + 𝑑2) (𝑒2 + 𝑓2) (𝑔2 + ℎ2) = 𝐴2 +𝐵2. Introduction (𝐴 + 𝑖𝐵) ( 𝐴 – 𝑖𝐵) Using ( a – b ) ( a + b ) = a2 – b2 = 𝐴2 – (𝑖𝐵)2 = 𝐴2 – 𝑖2 𝐵2 Putting i2 = −1 = 𝐴2 – ( −1) 𝐵2 = 𝐴2 +𝐵2 Hence, (𝐴 + 𝑖𝐵) (𝐴 – 𝑖𝐵) = 𝐴2 +𝐵2 Misc, 19 If (𝑎+𝑖𝑏)(𝑐+𝑖𝑑)(𝑒+𝑖𝑓)(𝑔+𝑖ℎ)=𝐴+𝑖𝐵, then show that (𝑎2 + 𝑏2) (𝑐2 + 𝑑2) (𝑒2 + 𝑓2) (𝑔2 + ℎ2) = 𝐴2 +𝐵2. Given ( 𝐴 + 𝑖𝐵 ) = (𝑎 + 𝑖𝑏 ) ( 𝑐 + 𝑖𝑑 ) (𝑒 + 𝑖𝑓 ) ( 𝑔 + 𝑖ℎ ) To calculate ( 𝐴 – 𝑖𝐵 ) Replacing 𝑖 by –𝑖 in (1) (𝐴 −𝑖𝐵 ) = ( 𝑎 – 𝑖𝑏 ) ( 𝑐 – 𝑖𝑑 ) ( 𝑒 – 𝑖𝑓 ) ( 𝑔 – 𝑖ℎ ) Now, calculating (𝐴 + 𝑖𝐵) ( 𝐴 – 𝑖𝐵) (𝐴 + 𝑖𝐵) ( 𝐴 – 𝑖𝐵) = (𝑎 + 𝑖𝑏 )( 𝑐 + 𝑖𝑑 )(𝑒 + 𝑖𝑓 )( 𝑔 + 𝑖ℎ )(𝑎 − 𝑖𝑏 ) ( 𝑐 − 𝑖𝑑 ) (𝑒 − 𝑖𝑓 ) ( 𝑔 − 𝑖ℎ ) 𝐴2 + 𝐵^2= [( 𝑎+ 𝑖𝑏 )(𝑎 – 𝑖𝑏 )][(𝑐+ 𝑖𝑑)(𝑐 – 𝑖𝑑 )] [( 𝑒 + 𝑖𝑓) ( 𝑒 – 𝑖𝑓 )] [( 𝑔 + 𝑖ℎ ) ( 𝑔 – 𝑖ℎ)] 𝑈𝑠𝑖𝑛𝑔 ( 𝑥 – 𝑦 ) ( 𝑥 + 𝑦 ) = 𝑥2+𝑦2 = [(𝑎)^2 – (𝑖𝑏)2] [ 𝑐2 – ( 𝑖𝑑)^2] [𝑒2− (𝑖𝑓)^2 ] [𝑔2 – (− 𝑖ℎ)]2 = [ 𝑎2 − 𝑏2 𝑖2 ] [ 𝑐2 − 𝑖2 𝑑2 ] [ 𝑒2 − 𝑖2 𝑓2 ] [ 𝑔2 − 𝑖2 ℎ2 ] Putting i2 = −1 = [ 𝑎2– (−1)𝑏2] [ 𝑐2 – (−1) 𝑑 ] [ 𝑒2 – (−1) 𝑓)] [𝑔2 – (−1) ℎ2 ] = [ 𝑎2 + 𝑏2 ] [ 𝑐2 + 𝑑2 ] [ 𝑒2 + 𝑓2 ] [𝑔2 + ℎ2 ] Hence, (𝑎2 + 𝑏2) (𝑐2 + 𝑑2) (𝑒2 + 𝑓2) (𝑔2 + ℎ2) = 𝐴2 +𝐵2. Hence proved

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.