Chapter 8.4, Problem 18E

Solution

To find: The focus of the parabola for $\left(x,y\right)$ from $\left(u,v\right)$ when the rotation equations are $x=0.8u-0.6v$ and $y=0.6u+0.8v$ .

The parabola focus $\left(4,2\right)$ in $\left(x,y\right)$ coordinates is $\left(4,3\right)$ .

Given information:

The rotation equations are,

  $\begin{array}{c}x=0.8u-0.6v\\ y=0.6u+0.8v\end{array}$

The coordinate in $\left(u,v\right)$ is parabola focus $\left(4,2\right)$ .

Calculation:

Calculate the value of u and v .

  $\begin{array}{c}\left(4,2\right)=\left(u,v\right)\\ u=4\\ v=2\end{array}$

Substitute 4 for $u$ and 2 for $v$ in above equations to find the value of $x$ .

  $\begin{array}{c}x=0.8\left(4\right)-0.6\left(2\right)\\ =3.2-1.2\\ =2\end{array}$

Similarly, substitute 4 for $u$ and 2 for $v$ in above equations to find the value of $y$ .

  $\begin{array}{c}y=0.6\left(4\right)+0.8\left(2\right)\\ =2.4+1.6\\ =4\end{array}$

So, the required focus is $\left(x,y\right)=\left(2,4\right)$ .

Therefore, the parabola focus in $\left(x,y\right)$ coordinates is $\left(2,4\right)$ .