Question

if x+iy=a+ib/a-ib prove that xsqaure+ysquare =1

Answer 1

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