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Misc 18 - |1 - i|x = 2x, find non integral solutions - Proof- Solving

Misc 18 - Chapter 5 Class 11 Complex Numbers - Part 2

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Misc, 18 Find the number of non-zero integral solutions of the equation |1 – 𝑖|π‘₯ = 2π‘₯ . We need to find the value of x which should be an integer but not 0 Lets first find the value of |1 – 𝑖| 1 – 𝑖 Complex number is of the form x + iy Where π‘₯ = 1 𝑦 = βˆ’1 |1 – 𝑖| = √(π‘₯^2+𝑦2) = √((1)2+(βˆ’1)2) = √(1+1) = √2 Hence, |1 – 𝑖| = √2 Given |1 – 𝑖|π‘₯ = 2π‘₯ Putting |1 – 𝑖| = √2 (√2)π‘₯ = 2π‘₯ (√2)π‘₯ = (√2 Γ— √2)π‘₯ (√2)π‘₯ = (√2)^(2 Γ— π‘₯) (√2)π‘₯ = γ€–βˆš2γ€—^( 2π‘₯) Comparing Powers π‘₯ = 2π‘₯ π‘₯ βˆ’2π‘₯ = 0 βˆ’ π‘₯ = 0 π‘₯ = 0 Hence, There is no non-zero integral solution

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