Paragraph: Consider the two complex numbers z and w such that w= Z-1 Z+2 =a+b i , where a, b R . Question: Which of the following is the value of - b a , whenever it exists?

Step-by-step solution

(i) Consider z= CiS and a+i b= z-1 z+2 aligned & a+i b= Cis -1 Cis +2 = ( -1)+i ( +2)+i = (( -1)+i )(( +2)-i ) ( +2)^2+ ^2 & = (( -1)( +2)+ ^2 )+i (( +2) ^ -( -1) ) ^2 +4 +4+ ^2 & = ( ^2 + + ^2 -2 )+i(3 ) 4 +5 = -1 4 +5 +i 3 4 +5 & a= -1 4 +5 ; b= 3 4 +5 ....(1) & 4 a +5 a= -1 (4 a-1) =-(1+5 a) or = 1+5 a 1-4 a ....(2) aligned Also, 4 ~b +5 b=3 . i.e. 3 = 4 b(1+5 a) (1-4 a) +5 b= 4 b+20 a b+5 b-20 a b (1-4 a) 3 = 9 b 1-4 a or = 3 b 1-4 a ....(3) as, ^2 + ^2 =1 aligned & 9 b^2 (1-4 a)^2 + (1+5 a)^2 (1-4 a)^2 =1 & i.