Misc 4 - If x - iy = root (a - ib)/(c - id), prove (x2 + y2)2 - Miscellaneous

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Misc 4 If x โ€“ iy = โˆš((a โˆ’ ib)/(c โˆ’ id)) prove that (๐‘ฅ2 + ๐‘ฆ2)^2 = (a^2 + b^2)/(c^2 + d^2 ) Introduction (๐‘ฅ โ€“ ๐‘–๐‘ฆ) (๐‘ฅ+ ๐‘–๐‘ฆ) Using ( a โ€“ b ) ( a + b ) = a2 โ€“ b2 = (๐‘ฅ)^2 โ€“ (๐‘–๐‘ฆ)2 = ๐‘ฅ2 โ€“ (๐‘–) 2๐‘ฆ2 = ๐‘ฅ2 โ€“ (โˆ’ 1)๐‘ฆ2 = ๐‘ฅ2 + ๐‘ฆ2 Misc 4 If x โ€“ iy = โˆš((a โˆ’ ib)/(c โˆ’ id)) prove that (๐‘ฅ2 + ๐‘ฆ2)^2 = (a^2 + b^2)/(c^2 + d^2 ) Given ๐‘ฅ โ€“ ๐‘–๐‘ฆ = โˆš((a โˆ’ ib)/(c โˆ’ id)) Calculating ๐‘ฅ + ๐‘–๐‘ฆ Replacing โ€“ ๐‘– by ๐‘– ๐‘ฅ + ๐‘–๐‘ฆ = โˆš((a + ib)/( c + id)) Multiplying (1) &(2) (๐‘ฅ โ€“๐‘–๐‘ฆ) (๐‘ฅ+ ๐‘–๐‘ฆ) = โˆš((a โˆ’ ib)/(c โˆ’ id)) ร— โˆš((a + ib)/(c + id)) ๐‘ฅ2+๐‘ฆ2 =โˆš((aโˆ’ib)/(cโˆ’id)ร—(a + ib)/(c + id)) =โˆš((( a โˆ’ ib) (a + ib))/((c โˆ’ id) (c + id))) Using ( a โ€“ b ) ( a + b ) = a2 โ€“ b2 =โˆš(((a)^2 โˆ’ (ib)^2 )/((c)^2โˆ’ใ€– (id)ใ€—^2 )) =โˆš((a^2 โˆ’ i^2 b^2 )/(c^2 โˆ’ i^2 d^2 )) Putting i2 = โˆ’1 =โˆš((a2โˆ’(โˆ’1) b2 )/(c2โˆ’(โˆ’1)d2)) =โˆš((a2+ b2 )/(c + d2)) Hence, ๐‘ฅ2 + ๐‘ฆ2 =โˆš((a2+ b2 )/(c2 + d2)) Squaring both sides (x2 + y2)2 =(โˆš((a2+ b2 )/(c2 + d2)))^2 (x2 + y2)2 = (a2+ b2 )/(c2 + d2) Hence Proved

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.