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

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Misc 4 - Chapter 5 Class 11 Complex Numbers - Part 2
Misc 4 - Chapter 5 Class 11 Complex Numbers - Part 3
Misc 4 - Chapter 5 Class 11 Complex Numbers - Part 4

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Misc 4 If x – iy = √((a βˆ’ ib)/(c βˆ’ id)) prove that (π‘₯2 + 𝑦2)^2 = (a^2 + b^2)/(c^2 + d^2 ) Introduction (π‘₯ – 𝑖𝑦) (π‘₯+ 𝑖𝑦) Using ( a – b ) ( a + b ) = a2 – b2 = (π‘₯)^2 – (𝑖𝑦)2 = π‘₯2 – (𝑖) 2𝑦2 = π‘₯2 – (βˆ’ 1)𝑦2 = π‘₯2 + 𝑦2 Misc 4 If x – iy = √((a βˆ’ ib)/(c βˆ’ id)) prove that (π‘₯2 + 𝑦2)^2 = (a^2 + b^2)/(c^2 + d^2 ) Given π‘₯ – 𝑖𝑦 = √((a βˆ’ ib)/(c βˆ’ id)) Calculating π‘₯ + 𝑖𝑦 Replacing – 𝑖 by 𝑖 π‘₯ + 𝑖𝑦 = √((a + ib)/( c + id)) Multiplying (1) &(2) (π‘₯ –𝑖𝑦) (π‘₯+ 𝑖𝑦) = √((a βˆ’ ib)/(c βˆ’ id)) Γ— √((a + ib)/(c + id)) π‘₯2+𝑦2 =√((aβˆ’ib)/(cβˆ’id)Γ—(a + ib)/(c + id)) =√((( a βˆ’ ib) (a + ib))/((c βˆ’ id) (c + id))) Using ( a – b ) ( a + b ) = a2 – b2 =√(((a)^2 βˆ’ (ib)^2 )/((c)^2βˆ’γ€– (id)γ€—^2 )) =√((a^2 βˆ’ i^2 b^2 )/(c^2 βˆ’ i^2 d^2 )) Putting i2 = βˆ’1 =√((a2βˆ’(βˆ’1) b2 )/(c2βˆ’(βˆ’1)d2)) =√((a2+ b2 )/(c + d2)) Hence, π‘₯2 + 𝑦2 =√((a2+ b2 )/(c2 + d2)) Squaring both sides (x2 + y2)2 =(√((a2+ b2 )/(c2 + d2)))^2 (x2 + y2)2 = (a2+ b2 )/(c2 + d2) 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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.