Question

X+iy=a+ib/a-ib,prove that x square+ysquare=1

Answer 1

question should be prove that x² -y²=1

first rationalize RHS ,
x+iy=[(a+ib)/(a-ib)]*[(a+ib)/(a+ib)]
x+iy= (a+ib)²/(a²+b²)
x+iy=(a²-b²+2abi)/(a²+b²)..........(1)
take cojugate
x-iy=(a²-b²-2abi)/(a²+b²)..........(2)

multiply (1) abd (2)
x²-y²=[(a² -b² )²+4a²b²]/(a²+b²)²
=[(a²)²+(b²)²+2a²b²]/(a²+b²)=(a²+b²)²/(a²+b²)²=1