Chapter 10.4, Problem 4AYU

To determine

Whether the equation $y²=9+x²$ is symmetric with respect to the $x\text{-axis}$, the $y\text{-axis}$, and the origin is true or false.

Answer to Problem 4AYU

Solution:

$\left(2,\text{ 5}\right)$

Explanation of Solution

Given:

The points $y²=9+x²$.

Formula used:

A graph is said be symmetric with respect to $y\text{-axis}$ if, $f\left(x,y\right)$ on the graph, the point $\left(-x,y\right)$ is also on the graph.

A graph is said be symmetric with respect to $x\text{-axis}$ if, $f\left(x,y\right)$ on the graph, the point $\left(x,-y\right)$ is also on the graph.

A graph is said be symmetric with respect to origin if, $f\left(x,y\right)$ on the graph, the point $\left(-x,-y\right)$ is also on the graph.

Calculation:

To check whether $y²=9+x²$ is symmetric with respect to $y\text{-axis}$,

Replacing $x$ by $-x$ we get,

$y²=9+{\left(-x\right)}^2=9+x²$

Hence, the equation is symmetric with respect to $y\text{-axis}$.

To check whether $y²=9+x²$ is symmetric with respect to $x\text{-axis}$,

Replacing $y$ by $-y$ we get,

${\left(-y\right)}^2=9+x^2;y²=9+x²$

Hence, the equation is symmetric with respect to $x\text{-axis}$.

To check whether $y²=9+x²$ is symmetric with respect to origin,

Replacing $x$ by $-x$ and $y$ by $-y$ we get,

${\left(-y\right)}^2=9+\text{ (}-x^2);y²=9+x²$

Hence, the equation is symmetric with respect to origin.