If z=x+i y, z^ 1 / 3 =a-i b , then x a - y b =k (a^2-b^2 ) , where k is equal to:

Step-by-step solution

aligned & Here, z^ 1 / 3 =a-i b & z=(a-i b)^3 & z=a^3-3 a^2(i b)+3 a(i b)^2-(i b)^3 & x+i y= (a^3-3 b^2 a )+i (-3 a^2 b+b^3 ) aligned Equating real and imaginary parts, we get x=a (a^2-3 b^2 ) and y=b (b^2-3 a^2 ) Now, a (a^2-3 b^2 ) a - b (b^2-3 a^2 ) b =k (a^2-b^2 ) 4 a^2-4 b^2=k (a^2-b^2 ) k=4