Answer
The solution is $\left\{ \left( 1,4 \right),\left( -1,-4 \right),\left( 2\sqrt{2},\sqrt{2} \right),\left( -2\sqrt{2},-\sqrt{2} \right) \right\}$
Work Step by Step
The given equations are,
$2{{x}^{2}}+{{y}^{2}}=18$ (I)
And
$ xy=4$ (II)
From equation (II):
$ y=\frac{4}{x}$
Substitute $ y=\frac{4}{x}$ in equation (I):
$\begin{align}
& 2{{x}^{2}}-{{\left( \frac{4}{x} \right)}^{2}}=18 \\
& {{x}^{4}}-9{{x}^{2}}+8=0 \\
& \left( {{x}^{2}}-8 \right)\left( {{x}^{2}}-1 \right)=0 \\
& x=\pm 2\sqrt{2},\pm 1\,
\end{align}$
Substitute $ x=2\sqrt{2}$ in equation (II):
$\begin{align}
& 2\sqrt{2}y=4 \\
& y=\sqrt{2}
\end{align}$
Substitute $ x=-2\sqrt{2}$ in equation (II):
$\begin{align}
& -2\sqrt{2}y=4 \\
& y=-\sqrt{2}
\end{align}$
Substitute $ x=1$ in equation (II):
$ y=4$
Substitute $ x=-1$ in equation (II):
$ y=-4$
Thus, the solution is,
$\left\{ \left( 1,4 \right),\left( -1,-4 \right),\left( 2\sqrt{2},\sqrt{2} \right),\left( -2\sqrt{2},-\sqrt{2} \right) \right\}$.
