Question

if x + yi = u+ vi / u-vi , prove that x² + y² =1​

Answer 1

Answer:

X+iy=a+ib/a-ib

or, x+iy=(a+ib)(a+ib)/(a-ib)(a+ib)

or, x+iy=(a²+2aib+i²b²)/(a²-i²b²)

or, x+iy={(a²-b²)+i.2ab}/(a²+b²)

or, x+iy=(a²-b²)/(a²+b²)+i. 2ab/(a²+b²)

∴, equating both sides, x=a²-b²/a²+b² and y=2ab/a²+b²

∴, x²+y²

= {(a²-b²)/(a²+b²)}²+{2ab/(a²+b²)}²

= {(a²-b²)²+4a²b²}/(a²+b²)²

= (a⁴-2a²b²+b⁴+4a²b²)/(a²+b²)²

= (a⁴+2a²b²+b⁴)/(a²+b²)²

= (a²+b²)²/(a²+b²)²

= 1

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