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

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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.