Answer
The given equation has symmetry about the x-axis, y-axis, and the origin also.
Work Step by Step
Step I: To check symmetry about the y-axis:
Put $x=-x$ in the given equation; if the equation remains the same, then it has symmetry about the y-axis.
$\begin{align}
& {{x}^{2}}+{{y}^{2}}=17 \\
& {{\left( -x \right)}^{2}}+{{y}^{2}}=17 \\
& {{x}^{2}}+{{y}^{2}}=17
\end{align}$
It is the same as the provided equation, hence it has symmetry about the y-axis.
Step II: To check symmetry about the x-axis:
Put $y=-y$ in the given equation; if the equation remains the same, then it has symmetry about the x-axis.
$\begin{align}
& {{x}^{2}}+{{y}^{2}}=17 \\
& {{x}^{2}}+{{\left( -y \right)}^{2}}=17 \\
& {{x}^{2}}+{{y}^{2}}=17
\end{align}$
It is the same as the provided equation, hence it has symmetry about the x-axis.
Step III: To check symmetry about the origin:
Put $x=-x\text{ and }y=-y$ in the given equation; if the equation remains the same, then it has symmetry about the origin.
$\begin{align}
& {{x}^{2}}+{{y}^{2}}=17 \\
& {{\left( -x \right)}^{2}}+{{\left( -y \right)}^{2}}=17 \\
& {{x}^{2}}+{{y}^{2}}=17
\end{align}$
It is the same as the provided equation, hence it has symmetry about the origin.
Therefore, the equation ${{x}^{2}}+{{y}^{2}}=17$ has symmetry about the x-axis, y-axis, and the origin also.
