Final answer:
The equation xy = 5 is symmetric with respect to the origin because any point (x, y) satisfying the equation will also have its reflected point (-x, -y) satisfy the equation.
Step-by-step explanation:
The correct answer is option C, Symmetric to the origin. To determine symmetry, we can perform tests for reflection across the x-axis, y-axis, and origin. For a graph to be symmetric about the x-axis, if any point (x, y) is on the graph, then (x, -y) must also be on the graph. Similarly, for y-axis symmetry, if (x, y) is on the graph, then (-x, y) must be on the graph. And for origin symmetry, if (x, y) is on the graph, then (-x, -y) must be on the graph.
To test the equation xy = 5 for symmetry, replace x with (-x) and y with (-y). If xy = 5 becomes (-x)(-y) = 5, the equation is unchanged because a negative times a negative is a positive. Thus, the equation is symmetric about the origin since for any point (x, y) that satisfies the equation, the point (-x, -y) will also satisfy it.
This can also be visualized by noting that reflecting a point on the hyperbola xy = 5 across either the x-axis or the y-axis and then across the other axis leads back to the hyperbola, showing origin symmetry.