Question
Let $$x_1$$ and $$y_1$$ be real numbers. If $$z_1$$ and $$z_2$$ are complex numbers such that $$\left| {{z_1}} \right| = \left| {{z_2}} \right| = 4,$$ then $${\left| {{x_1}{z_1} - {y_1}{z_2}} \right|^2} + {\left| {{y_1}{z_1} + {x_1}{z_2}} \right|^2} =? $$
A.
$$32\left( {{x_1}^2 + {y_1}^2} \right)$$
B.
$$16\left( {{x_1}^2 + {y_1}^2} \right)$$
C.
$$4\left( {{x_1}^2 + {y_1}^2} \right)$$
D.
$$32\left( {{x_1}^2 + {y_1}^2} \right){\left| {{z_1} + {z_2}} \right|^2}$$
Answer :
$$32\left( {{x_1}^2 + {y_1}^2} \right)$$
Solution :
$$\eqalign{
& {\left| {{x_1}{z_1} - {y_1}{z_2}} \right|^2} + {\left| {{y_1}{z_1} - {x_1}{z_2}} \right|^2} \cr
& = {\left| {{x_1}{z_1}} \right|^2} + {\left| {{y_1}{z_2}} \right|^2} - 2\operatorname{Re} \left( {{x_1}{y_1}{z_1}{z_2}} \right) + {\left| {{y_1}{z_1}} \right|^2} + {\left| {{x_1}{z_2}} \right|^2} + 2\operatorname{Re} \left( {{x_1}{y_1}{z_1}{z_2}} \right) \cr
& = {x_1}^2{\left| {{z_1}} \right|^2} + {y_1}^2{\left| {{z_2}} \right|^2} + {y_1}^2{\left| {{z_1}} \right|^2} + {x_1}^2{\left| {{z_2}} \right|^2} \cr
& = 2\left( {{x_1}^2 + {y_1}^2} \right)\left( {{4^2}} \right) = 32\left( {{x_1}^2 + {y_1}^2} \right) \cr} $$
Alternate Solution
$$\eqalign{
& {\text{Given, }}\left| {{z_1}} \right| = \left| {{z_2}} \right| = 4......\left( {\text{i}} \right) \cr
& {\text{Now, }}{\left| {{x_1}{z_1} - {y_1}{z_2}} \right|^2} + {\left| {{y_1}{z_1} - {x_1}{z_2}} \right|^2} \cr
& {\text{ = }}{\left| {{x_1}{z_1}} \right|^2} + {\left| {{y_1}{z_2}} \right|^2} - 2{x_1}{z_1}.{y_1}{z_2} + {\left| {{y_1}{z_1}} \right|^2} + {\left| {{x_1}{z_2}} \right|^2} + 2{y_1}{z_1}{x_1}{z_2} \cr
& {\text{ = }}{\left| {{x_1}} \right|^2}{\left| {{z_1}} \right|^2} + {\left| {{y_1}} \right|^2}{\left| {{z_2}} \right|^2} + {\left| {{y_1}} \right|^2}{\left| {{z_1}} \right|^2} + {\left| {{x_1}} \right|^2}{\left| {{z_2}} \right|^2} \cr
& = x_1^2{\left( 4 \right)^2} + y_1^2{\left( 4 \right)^2} + y_1^2{\left( 4 \right)^2} + x_1^2{\left( 4 \right)^2}\,\,\,\,\,\,\left( {\because \,{{\left| z \right|}^2} = {z^2}} \right) \cr
& {\text{ = 2}}{\text{.}}{\left( 4 \right)^2}.x_1^2 + 2.{\left( 4 \right)^2}.y_1^2 \cr
& {\text{ = 32}}\left( {x_1^2 + y_1^2} \right) \cr} $$