Chapter 3.1, Problem 38E

(a)

To determine

To find: The symmetry in the graph of the comet’s path.

Answer to Problem 38E

The explanation is given below.

Explanation of Solution

Given:

The path of a comet around the sun can be modelled by a transformation of the equation $\frac{x^2}{8}+\frac{y^2}{10}=1$ .

Let us check the symmetry of graph at $x$ -axis as below:

  $\begin{array}{c}\frac{x^2}{8}+\frac{{\left(-y\right)}^2}{10}=1\\ \frac{x^2}{8}+\frac{{\left(y\right)}^2}{10}=1\end{array}$

Let us check the symmetry of graph at $y$ -axis as below:

  $\begin{array}{c}\frac{{\left(-x\right)}^2}{8}+\frac{{\left(y\right)}^2}{10}=1\\ \frac{x^2}{8}+\frac{{\left(y\right)}^2}{10}=1\end{array}$

Let us check the symmetry of graph first at origin as below:

  $\begin{array}{c}\frac{{\left(-x\right)}^2}{8}+\frac{{\left(-y\right)}^2}{10}=1\\ \frac{x^2}{8}+\frac{{\left(y\right)}^2}{10}=1\end{array}$

The graph is symmetric to all three.

(b)

To determine

To find: The graph of the equation $\frac{x^2}{8}+\frac{y^2}{10}=1$ using symmetry.

Answer to Problem 38E

The graph is given below.

Explanation of Solution

The graph by using symmetry is given below:

  

Therefore the graph is given above.

(c)

To determine

To find: The name of the coordinates of three other points through which the comet pass.

Answer to Problem 38E

The coordinates are $\left(-2,\sqrt{5}\right),\left(-2,-\sqrt{5}\right),\text{and}\left(2,-\sqrt{5}\right)$ .

Explanation of Solution

Given:

The comet passes through the point at $\left(2,\sqrt{5}\right)$

As given that the comet passes through the point at $\left(2,\sqrt{5}\right)$ and the path of comet is symmetry. So if the comet passes through $\left(2,\sqrt{5}\right)$ so it must also pass through the points $\left(-2,\sqrt{5}\right),\left(-2,-\sqrt{5}\right),\text{and}\left(2,-\sqrt{5}\right)$

Therefore the coordinates are $\left(-2,\sqrt{5}\right),\left(-2,-\sqrt{5}\right),\text{and}\left(2,-\sqrt{5}\right)$ .