If a, b R and i= -1 , then the number of ordered pairs of real numbers (a, b) satisfying the condition (a+b i)^3=a-b i is

Step-by-step solution

Given condition, aligned & (a+b i)^3=a-b i & i.e. a^3+i^3 b^3+3 a^2 b i-3 a b^2 i^2=a-b i & a^3-i b^3+i 3 a^2 b-3 a b^2=a-b i & (a^3-3 a b^2 )-i (b^3-3 a^2 b )=a-b i & aligned Equating the coefficient, we obtain aligned a^3-3 a b^2 & =a and b^3-3 a^2 b=b a (a^2-3 b^2 ) & =a and b (b^2-3 a^2 )=b & a^2-3 b^2=1 and b^2-3 a^2 & =1 aligned Use Eq. (i) in (ii) aligned b^2-3 (1+3 b^2 ) & =1 b^2-3-9 b^2 & =1 or 8 b^2+4=0 aligned This gives b^2=-1 / 2 b= i 2 Now, aligned a^2 & =1+3 b^2=1+3 ( -1 2 )=1- 3 2 =- 1 2 a & = i 2 a