Answer
(a) This graph is not symmetric about the $y$-axis, is symmetric about the $x$-axis, and is not symmetric about the origin.
(b) This graph is symmetric about the $y$-axis, is symmetric about the $x$-axis, and is symmetric about the origin.
(c) This graph is not symmetric about the $y$-axis, is not symmetric about the $x$-axis, and is symmetric about the origin.
Work Step by Step
(a) For the graph of $x=5y^2+9$ we have
(i) $$x \to -x \quad \Rightarrow \quad (x=5y^2+9) \to (-x=5y^2+9),$$so it is not symmetric about the $y$-axis,
(ii) $$y \to -y \quad \Rightarrow \quad (x=5y^2+9) \to (x=5(-y)^2+9=5y^2+9),$$so it is symmetric about the $x$-axis,
(iii) $$x \to -x, \quad y \to -y \quad \Rightarrow \quad (x=5y^2+9) \to (-x=5(-y)^2+9=5y^2+9),$$so it is not symmetric about the origin.
(b) For the graph of $x^2-2y^2=3$ we have
(i) $$x \to -x \quad \Rightarrow \quad (x^2-2y^2=3) \to ((-x)^2-2y^2=x^2-2y^2=3),$$so it is symmetric about the $y$-axis,
(ii) $$y \to -y \quad \Rightarrow \quad (x^2-2y^2=3) \to (x^2-2(-y)^2=x^2-2y^2=3),$$so it is symmetric about the $x$-axis,
(iii) $$x \to -x, \quad y \to -y \quad \Rightarrow \quad (x^2-2y^2=3) \to ((-x)^2-2(-y)^2=x^2-2y^2=3),$$so it is symmetric about the origin.
(c) For the graph of $xy=5$ we have
(i) $$x \to -x \quad \Rightarrow \quad (xy=5) \to (-xy=5),$$ so it is not symmetric about the $y$-axis,
(ii) $$y \to -y \quad \Rightarrow \quad (xy=5) \to (-xy=5),$$so it is not symmetric about the $x$-axis,
(iii) $$x \to -x, \quad y \to -y \quad \Rightarrow \quad (xy=5) \to ((-x)(-y)=xy=5),$$so it is symmetric about the origin.
