Chapter 1, Problem 11RE

In Problems 10-14, test each equation for symmetry with respect to the $x-axis$, the $y-axis$, and the origin.

$x^2+4y^2=16$

To determine

To test: The equation $x²+4y²=16$ for symmetry, with respect to the $x\text{-axis}$, $y\text{-axis}$ and the origin.

Answer to Problem 11RE

Solution:

$\text{The graph of the given equation is symmetric, with respect to }x\text{-axis, }y\text{-axis and the origin}$.

Given:

The equation $x²+4y²=16$

Explanation of Solution

Replace $y$ by $-y$ in the equation $x²+4y²=16$.

$x²+4\left(-y\right)²=16$

$x²+4y²=16$, which is equivalent to the given equation.

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

Replace $x$ by $-x$

$\left(-x\right)²+4y²=16$

$x²+4y²=16$, which is equivalent to the given equation.

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

Replace $x$ by $-x$ and $y$ by $-y$.

$\left(-x\right)²+4\left(-y\right)²=16$

$x²+4y²=16$, which is equivalent to the given equation.

Hence the graph is symmetric, with respect to the origin.