Question

If a + ib = , prove that a 2 + b 2 =

Answer 1

The given expression is,

a+ib= ( x+i ) 2 2 x 2 +1 = x 2 + i 2 +2xi 2 x 2 +1 = x 2 −1+2xi 2 x 2 +1 = x 2 −1 2 x 2 +1 +i 2x 2 x 2 +1

Comparing the real and imaginary parts, we get

a= x 2 −1 2 x 2 +1 b= 2x 2 x 2 +1

Therefore,

a 2 + b 2 = ( x 2 −1 2 x 2 +1 ) 2 + ( 2x 2 x 2 +1 ) 2 = x 4 +1−2 x 2 +4 x 2 ( 2 x 2 +1 ) 2 = x 4 +1+2 x 2 ( 2 x 2 +1 ) 2 = ( x 2 +1 ) 2 ( 2 x 2 +1 ) 2

Hence, proved.