Chapter 1.8, Problem 79E

To determine

To calculate:Whether the equation $x^2{y}^2+xy=1$ is symmetric about x-axis or y-axis or origin.

Answer to Problem 79E

The equation $x^2{y}^2+xy=1$ is symmetric only with respect to origin.

Explanation of Solution

Given information:

The equation $x^2{y}^2+xy=1$ .

Formula used:

The function is symmetric about the x-axis, when y is replaced by $-y$ , the equation remains unchanged.

The function is symmetric about the y-axis, when x is replaced by $-x$ , the equation remains unchanged.

The function is symmetric with respect to origin, when y is replaced by $-y$ and x is replaced by $-x$ , the equation remains unchanged.

Calculation:

It is provided that the equation is $x^2{y}^2+xy=1$ . Construct a table to evaluate the value of y for different values of x.

Recall that the function is symmetric about the x-axis, when y is replaced by $-y$ , the equation remains unchanged.

Replace y by $-y$ in the equation $x^2{y}^2+xy=1$ ,

  $\begin{array}{c}{x}^2{\left(-y\right)}^2+x\left(-y\right)=1\\ x^2{y}^2-xy=1\end{array}$

The equation is changed. Therefore, the equation $x^2{y}^2+xy=1$ is not symmetricabout the x-axis.

Recall that the function is symmetric about the y-axis, when x is replaced by $-x$ , the equation remains unchanged.

Replace x by $-x$ in the equation $x^2{y}^2+xy=1$ ,

  $\begin{array}{c}{\left(-x\right)}^2{y}^2+\left(-x\right)y=1\\ x^2{y}^2-xy=1\end{array}$

The equation is changed. Therefore, the equation $x^2{y}^2+xy=1$ is not symmetricabout the y-axis.

Recall that the function is symmetric with respect to origin, when y is replaced by $-y$ and x is replaced by $-x$ , the equation remains unchanged.

Replace x by $-x$ and y by $-y$ in the equation $x^2{y}^2+xy=1$ ,

  $\begin{array}{c}{\left(-x\right)}^2{y}^2+\left(-x\right)\left(-y\right)=1\\ x^2{y}^2+xy=1\end{array}$

The equation is unchanged. Therefore, the equation $x^2{y}^2+xy=1$ is symmetricabout the origin.

Thus, the equation is symmetric only about origin.