Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$x y = 11$$
$2 y ^ { 2 } - x ^ { 2 } = 11$
A) $\left\{ \left( \mathrm { i } \sqrt { 22 } , - \frac { \mathrm { i } \sqrt { 22 } } { 2 } \right) , ( \sqrt { 11 } , \sqrt { 11 } ) \right\}$
B) $\left\{ \left( - 22 , - \frac { 1 } { 2 } \right) , ( 11,1 ) \right\}$
C) $\{ ( 2 \mathrm { i } \sqrt { 11 } , - \mathrm { i } \sqrt { 11 } ) , ( - 2 \mathrm { i } \sqrt { 11 } , \mathrm { i } \sqrt { 11 } ) , ( \sqrt { 11 } , \sqrt { 11 } ) , ( - \sqrt { 11 } , - \sqrt { 11 } ) \}$
D) $\left\{ \left( \mathrm { i } \sqrt { 22 } , - \frac { \mathrm { i } \sqrt { 22 } } { 2 } \right) , \left( - \mathrm { i } \sqrt { 22 } , \frac { \mathrm { i } \sqrt { 22 } } { 2 } \right) , ( \sqrt { 11 } , \sqrt { 11 } ) , ( - \sqrt { 11 } , - \sqrt { 11 } ) \right\}$

Question

Quizplus · Accepted Answer

The system of equations is nonlinear and involves both real and complex solutions. By solving the system, we find that it has four pairs of solutions: two pairs involving real numbers $(\sqrt{11}, \sqrt{11})$ and $(- \sqrt{11}, - \sqrt{11})$, and two pairs involving complex numbers $\left( \mathrm { i } \sqrt { 22 } , - \frac { \mathrm { i } \sqrt { 22 } } { 2 } \right)$ and $\left( - \mathrm { i } \sqrt { 22 } , \frac { \mathrm { i } \sqrt { 22 } } { 2 } \right)$. These solutions satisfy both equations in the system, taking into account the properties of complex numbers and the relationship between $x$ and $y$ in the given equations.

Give All Solutions of the Nonlinear System of Equations, Including xy=11x y = 11xy=11

Give All Solutions of the Nonlinear System of Equations, Including $x y = 11$