Miscellaneous

Chapter 5 Class 11 Complex Numbers
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Misc 4 If x – iy = √((a − ib)/(c − id)) prove that (𝑥2 + 𝑦2)^2 = (a^2 + b^2)/(c^2 + d^2 ) Introduction (𝑥 – 𝑖𝑦) (𝑥+ 𝑖𝑦) Using ( a – b ) ( a + b ) = a2 – b2 = (𝑥)^2 – (𝑖𝑦)2 = 𝑥2 – (𝑖) 2𝑦2 = 𝑥2 – (− 1)𝑦2 = 𝑥2 + 𝑦2 Misc 4 If x – iy = √((a − ib)/(c − id)) prove that (𝑥2 + 𝑦2)^2 = (a^2 + b^2)/(c^2 + d^2 ) Given 𝑥 – 𝑖𝑦 = √((a − ib)/(c − id)) Calculating 𝑥 + 𝑖𝑦 Replacing – 𝑖 by 𝑖 𝑥 + 𝑖𝑦 = √((a + ib)/( c + id)) Multiplying (1) &(2) (𝑥 –𝑖𝑦) (𝑥+ 𝑖𝑦) = √((a − ib)/(c − id)) × √((a + ib)/(c + id)) 𝑥2+𝑦2 =√((a−ib)/(c−id)×(a + ib)/(c + id)) =√((( a − ib) (a + ib))/((c − id) (c + id))) Using ( a – b ) ( a + b ) = a2 – b2 =√(((a)^2 − (ib)^2 )/((c)^2−〖 (id)〗^2 )) =√((a^2 − i^2 b^2 )/(c^2 − i^2 d^2 )) Putting i2 = −1 =√((a2−(−1) b2 )/(c2−(−1)d2)) =√((a2+ b2 )/(c + d2)) Hence, 𝑥2 + 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 