Answer
The graph of the equation ${{x}^{2}}{{y}^{2}}+3xy=1$ symmetric to the origin.
Work Step by Step
Consider the equation, ${{x}^{2}}{{y}^{2}}+3xy=1$
Check symmetry about the y-axis:
An equation is symmetric about the y-axis if $-x$ is substituted in the function and the result is an equivalent equation then the graph of the equation is symmetric with respect to the y-axis.
Substitute $x=-x$ in the equation,
$\begin{align}
& {{\left( -x \right)}^{2}}{{y}^{2}}+3\left( -x \right)y=1 \\
& {{x}^{2}}{{y}^{2}}-3xy=1
\end{align}$
Therefore, the equation is not symmetric about the y-axis
Now, check symmetry about the x-axis:
An equation is symmetric about the x-axis, if $-y$ is substituted in the function and it leads to an equivalent equation than the graph is symmetric with respect to the x-axis.
Substitute $y=-y$ in the equation ${{x}^{2}}{{y}^{2}}+3xy=1$.
$\begin{align}
& {{x}^{2}}{{\left( -y \right)}^{2}}+3x\left( -y \right)=1 \\
& {{x}^{2}}{{y}^{2}}-3xy=1
\end{align}$
Therefore, the equation is not symmetric about the x-axis.
Now, check symmetry about the origin:
An equation is symmetric about origin if $x=-x,y=-y$ and it leads to an equivalent equation, this implies that the function is symmetric about the origin.
Substitute $x=-x,y=-y$ in the equation ${{x}^{2}}{{y}^{2}}+3xy=1$.
$\begin{align}
& {{\left( -x \right)}^{2}}{{\left( -y \right)}^{2}}+3\left( -x \right)\left( -y \right)=1 \\
& {{x}^{2}}{{y}^{2}}+3xy=1
\end{align}$
Therefore, the equation is symmetric about the origin.
Hence, the equation is symmetric about the origin.
