Question

determine whether the graph of the equation is symmetric with respect to the y - axis, the x - axis, the origin, more than one of these, or none of these.
$x^{2}y^{2}+5xy = 1$
select all that apply
□ none of these
□ y - axis
□ x - axis
□ origin

Explanation:

Step1: Test y-axis symmetry

Replace $x$ with $-x$:
$$(-x)^2y^2 + 5(-x)y = x^2y^2 - 5xy$$
This is not equal to the original equation $x^2y^2 + 5xy = 1$, so no y-axis symmetry.

Step2: Test x-axis symmetry

Replace $y$ with $-y$:
$$x^2(-y)^2 + 5x(-y) = x^2y^2 - 5xy$$
This is not equal to the original equation, so no x-axis symmetry.

Step3: Test origin symmetry

Replace $x$ with $-x$ and $y$ with $-y$:
$$(-x)^2(-y)^2 + 5(-x)(-y) = x^2y^2 + 5xy$$
This matches the original equation, so origin symmetry exists.

Answer:

origin