If a>0 and z= (1+i)^2 a+i ,(i= -1 ) has magnitude 2 5 , then z is equal to

Step-by-step solution

aligned & z= (1+i)^2 a+i & = 2 i a + i & = 2 i(a-i) (a+i)(a-i) & = 2+2 ai a ^2+1 & |z|= 2 5 4 (a^2+1 )^2 + 4 a^2 (a^2+1 )^2 = 4 5 & 20+20 a^2=4 (a^4+2 a^2+1 ) & 4 a^4-12 a^2-16=0 & a^4-3 a^2-4=0 & ( a ^2-4 ) ( a ^2+1 )=0 & a^2=4 and a^2=-1 & a =2 & [ a>0] & (i) z= 2 5 + 4 5 i & z = 2 5 - 4 5 i & aligned