## Precalculus (6th Edition) Blitzer

The graph of the equation ${{x}^{2}}{{y}^{2}}+5xy=2$ is symmetric to the origin.
Consider the equation, ${{x}^{2}}{{y}^{2}}+5xy=2$ 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}}+5\left( -x \right)y=2 \\ & {{x}^{2}}{{y}^{2}}-5xy=2 \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 ${{x}^{2}}{{y}^{2}}+5xy=2$ \begin{align} & {{x}^{2}}{{\left( -y \right)}^{2}}+5x\left( -y \right)=2 \\ & {{x}^{2}}{{y}^{2}}-5xy=2 \end{align} Therefore, the equation is not symmetric about the x-axis. Now, check symmetry about the origin: An equation is symmetric about the 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 ${{x}^{2}}{{y}^{2}}+5xy=2$. \begin{align} & {{\left( -x \right)}^{2}}{{\left( -y \right)}^{2}}+5\left( -x \right)\left( -y \right)=2 \\ & {{x}^{2}}{{y}^{2}}+5xy=2 \end{align} Therefore, the equation is symmetric about the origin.