Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$\begin{array} { r }
x y = 11 \
2 y ^ { 2 } - x ^ { 2 } = 11
\end{array}$$
A) $\left\{ \left( - 22 , - \frac { 1 } { 2 } \right) , ( 11,1 ) \right\}$

B) $\left\{ \left( i \sqrt { 22 } , - \frac { i \sqrt { 22 } } { 2 } \right) , \left( - i \sqrt { 22 } , \frac { i \sqrt { 22 } } { 2 } \right) , ( \sqrt { 11 } , \sqrt { 11 } ) , ( - \sqrt { 11 } , - \sqrt { 11 } ) \right\}$

C) $\left\{ \left\{ \mathrm { i } \sqrt { 22 } , - \frac { \mathrm { i } \sqrt { 22 } } { 2 } \right\} , ( \sqrt { 11 } , \sqrt { 11 } ) \right\}$

D) $\{ ( 2 \mathrm { i } \sqrt { 11 } , - \mathrm { i } \sqrt { 11 } ) , ( - 2 \mathrm { i } \sqrt { 11 } , \mathrm { i } \sqrt { 11 } ) , ( \sqrt { 11 } , \sqrt { 11 } ) , ( - \sqrt { 11 } , - \sqrt { 11 } ) \}$

Question

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$\begin{array} { r } 
x y = 11 \
2 y ^ { 2 } - x ^ { 2 } = 11
\end{array}$$
A) $\left\{ \left( - 22 , - \frac { 1 } { 2 } \right) , ( 11,1 ) \right\}$

B) $\left\{ \left( i \sqrt { 22 } , - \frac { i \sqrt { 22 } } { 2 } \right) , \left( - i \sqrt { 22 } , \frac { i \sqrt { 22 } } { 2 } \right) , ( \sqrt { 11 } , \sqrt { 11 } ) , ( - \sqrt { 11 } , - \sqrt { 11 } ) \right\}$

C) $\left\{ \left\{ \mathrm { i } \sqrt { 22 } , - \frac { \mathrm { i } \sqrt { 22 } } { 2 } \right\} , ( \sqrt { 11 } , \sqrt { 11 } ) \right\}$

D) $\{ ( 2 \mathrm { i } \sqrt { 11 } , - \mathrm { i } \sqrt { 11 } ) , ( - 2 \mathrm { i } \sqrt { 11 } , \mathrm { i } \sqrt { 11 } ) , ( \sqrt { 11 } , \sqrt { 11 } ) , ( - \sqrt { 11 } , - \sqrt { 11 } ) \}$

Quizplus · Accepted Answer

To solve the system, substitute $xy = 11$ into the second equation to get $2y^2 - (11/y)^2 = 11$. Solving this equation yields four solutions: $y = \sqrt{11}$, $y = -\sqrt{11}$, $y = -\frac{i\sqrt{22}}{2}$, and $y = \frac{i\sqrt{22}}{2}$. Substituting these values back into $xy = 11$ gives the corresponding $x$ values, resulting in the four pairs of solutions listed in option B.

Give All Solutions of the Nonlinear System of Equations, Including xy=112y2−x2=11\begin{array} { r } x y = 11 \\2 y ^ { 2 } - x ^ { 2 } = 11\end{array}xy=112y2−x2=11​

Give All Solutions of the Nonlinear System of Equations, Including $\begin{array} { r } x y = 11 \\2 y ^ { 2 } - x ^ { 2 } = 11\end{array}$