Problem 30 - Precalculus Essentials

Question

Determine whether the graph of each equation is symmetric with respect to the \(y\) -axis, the \(x\) -axis, the origin, more than one of these, or none of these. $$x^{2} y^{2}+5 x y=2$$

Step-by-Step Solution

Answer

The graph of the equation \(x^{2} y^{2}+5 x y=2\) is symmetric with respect to the origin.

Step 1: Checking symmetry with respect to the x-axis

To check for symmetry about the x-axis, replace \(y\) with \(-y\) in the equation \(x^{2} y^{2}+5 x y=2\). This gives \(x^{2} (-y)^{2}+5 x (-y)=2\), which simplifies to \(x^{2} y^{2}-5 x y=2\). This equation is different from the original equation, so the graph is not symmetric with respect to the x-axis.

Step 2: Checking symmetry with respect to the y-axis

To check for symmetry about the y-axis, replace \(x\) with \(-x\) in the equation \(x^{2} y^{2}+5 x y=2\). This gives \((-x)^{2} y^{2}+5 (-x) y=2\), which simplifies to \(x^{2} y^{2}-5 x y=2\). This equation is different from the original equation, so the graph is not symmetric with respect to the y-axis.

Step 3: Checking symmetry with respect to the origin

To check for symmetry about the origin, replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation \(x^{2} y^{2}+5 x y=2\). This gives \((-x)^{2} (-y)^{2}+5 (-x) (-y)=2\), which simplifies to \(x^{2} y^{2}+5 x y=2\). This equation is the same as the original equation, so the graph is symmetric with respect to the origin.