Chapter A5.3, Problem 42E

(a)

To determine

To find: The effect of rotating ${180}^{\circ }$ on the given equation.

Answer to Problem 42E

Here, the equation for the ellipse is $\frac{x{\prime }^2}{a^2}+\frac{y{\prime }^2}{b^2}=1$ .

Explanation of Solution

Given: $\alpha ={180}^{\circ }$ .

Calculation:

Given,

  $\alpha ={180}^{\circ }$

Therefore,

  $\begin{array}{l}\sin \alpha =0\\ \cos \alpha =-1\end{array}$

Now, after rotation of axes,

  $\begin{array}{l}x=x\prime \cos \alpha -y\prime \sin \alpha \\ =-x\prime \\ y=x\prime \sin \alpha -y\prime \cos \alpha \\ =-y\prime \end{array}$

Given equation is :

  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Substituting the values of $x=-x\prime$ and $y=-y\prime$ ,

  $\begin{array}{l}\frac{{\left(-x\prime \right)}^2}{a^2}+\frac{{\left(-y\prime \right)}^2}{b^2}=1\\ \Rightarrow \frac{x{\prime }^2}{a^2}+\frac{y{\prime }^2}{b^2}=1\end{array}$

Conclusion:

Hence, the equation for the ellipse is $\frac{x{\prime }^2}{a^2}+\frac{y{\prime }^2}{b^2}=1$ .

(b)

To determine

To find: The effect of rotating ${180}^{\circ }$ on the given equation.

Answer to Problem 42E

Here, the equation for the hyperbola is $\frac{x{\prime }^2}{a^2}-\frac{y{\prime }^2}{b^2}=1$ .

Explanation of Solution

Calculation:

Given equation is :

  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Substituting the values of $x=-x\prime$ and $y=-y\prime$ ,

  $\begin{array}{l}\frac{{\left(-x\prime \right)}^2}{a^2}-\frac{{\left(-y\prime \right)}^2}{b^2}=1\\ \Rightarrow \frac{x{\prime }^2}{a^2}-\frac{y{\prime }^2}{b^2}=1\end{array}$

Conclusion:

Hence, the equation for the hyperbola is $\frac{x{\prime }^2}{a^2}-\frac{y{\prime }^2}{b^2}=1$ .

(c)

To determine

To find: The effect of rotating ${180}^{\circ }$ on the given equation.

Answer to Problem 42E

Here, the equation for the circle is $x{\prime }^2+y{\prime }^2=a^2$ .

Explanation of Solution

Calculation:

Given equation is:

  $x^2+y^2=a^2$

Substituting the values of $x=-x\prime$ and $y=-y\prime$ ,

  $\begin{array}{l}{\left(-x\prime \right)}^2+{\left(-y\prime \right)}^2=a^2\\ \Rightarrow x{\prime }^2+y{\prime }^2=a^2\end{array}$

Conclusion:

Hence, the equation for the circle is $x{\prime }^2+y{\prime }^2=a^2$ .

(d)

To determine

To find: The effect of rotating ${180}^{\circ }$ on the given equation.

Answer to Problem 42E

Here, the equation for the line is $y\prime =mx\prime$ .

Explanation of Solution

Calculation:

Given equation is :

  $y=mx$

Substituting the values of $x=-x\prime$ and $y=-y\prime$ ,

  $\begin{array}{l}\left(-y\prime \right)=m\left(-x\prime \right)\\ \Rightarrow y\prime =mx\prime \end{array}$

Conclusion:

Hence, the equation for the line is $y\prime =mx\prime$ .

(e)

To determine

To find: The effect of rotating ${180}^{\circ }$ on the given equation.

Answer to Problem 42E

Here, the equation for the line is $y\prime =mx\prime -b$ .

Explanation of Solution

Calculation:

Given equation is:

  $y=mx+b$

Substituting the values of $x=-x\prime$ and $y=-y\prime$ ,

  $\begin{array}{l}\left(-y\prime \right)=m\left(-x\prime \right)+b\\ \Rightarrow y\prime =mx\prime -b\end{array}$

Conclusion:

Hence, the equation for the line is $y\prime =mx\prime -b$ .