Question

\(If (a+ib) (c+id) (e+if)(g+ih)=A+iB, \text {then show that} (a^2+b^2 )(c^2+d^2) (e^2+f^2)(g^2+h^2)=A^2+B^2.\)

Answer 1

\((a+ib)(c+id)(e+if)(g+ih)=A+iB\)

\(∴|(a+ib)(c+id)(e+if)(g+ih)|=|A+iB|\)

\(⇒|(a+ib)|×|(c+id)|×|(e+if)|×|(g+ih)|=|A+iB|\)          \([|z_1z_2|=|z_1||z_2|]\)

\(⇒ \sqrt a^2+b^2×\sqrt c^2+d^2×\sqrt e^2+f^2×\sqrt g^2+h^2=\sqrt A^2+B^2\)

\(\text{On squaring both sides, we obtain}\)

\((a^ 2  + b^ 2 ) (c^ 2  + d ^2 ) (e ^2  + f ^2 ) (g^ 2  + h^ 2 ) = A^2 + B ^2\)

\(\text{Hence, proved.}\)