Chapter 1.4, Problem 44E

(a)

To determine

To draw: Thegraph of equations $x=\left(2\cos t\right)+h$ and $y=\left(2\sin t\right)+k$ with the conditions $k=0$ and $h=-2,-1,1,2$ . Also, describe the role of h .

Explanation of Solution

Given information:The given parametric equations are $x=\left(2\cos t\right)+h$ and $y=\left(2\sin t\right)+k$ with the conditions $k=0$ and $h=-2,-1,1,2$ for the parametric interval $\left[0,2\pi \right]$ .

Graph:

To draw the graph of parametric equations $x=\left(2\cos t\right)+h$ and $y=\left(2\sin t\right)+k$ by the graphing calculator, follow the steps given below.

Step 1: Press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key to graph the functions. Now, press $\boxed{\text{MODE}}$ key to select Func, and then Par under the Normal to draw parametric equations.

Step 2: For $k=0$ and $h=-2$ , enter the keystrokes $\boxed{2}\boxed{\text{COS}}\boxed{\text{T}}\boxed{)}\boxed{-}\boxed{2}$ after the symbol ${\text{X}}_{\text{1T}}$ for the equation $x=2\cos t-2$ . Now, press $\boxed{\text{ENTER}}$ and enter the keystrokes $\boxed{2}\boxed{\text{SIN}}\boxed{\text{T}}\boxed{)}$ after the symbol ${\text{Y}}_{\text{1T}}$ for the equation $y=2\sin t$ .

Similarly, enter the equations for different values of $h=-1$ , $h=1$ and $h=2$ .

Step 3: Press $\boxed{\text{WINDOW}}$ key and set the parametric interval at $\left[0,2\pi \right]$ . Also, set the window at $\left[-6,6\right]$ by $\left[-4,4\right]$ . Press $\boxed{\text{GRAPH}}$ key to produce the graph of function as shown below.

  

Figure (1)

Interpretation: From the graph it can be observed that graph is a circle of radius 2 with center $\left(h,0\right)$ . The graph shifts horizontally as the values of $h$ changes.

(b)

To determine

To draw: The graph of equations $x=\left(2\cos t\right)+h$ and $y=\left(2\sin t\right)+k$ with the conditions $h=0$ and $k=-2,-1,1,2$ . Also, describe the role of k .

Explanation of Solution

Given information:The given parametric equations are $x=\left(2\cos t\right)+h$ and $y=\left(2\sin t\right)+k$ with the conditions $h=0$ and $k=-2,-1,1,2$ for the parametric interval $\left[0,2\pi \right]$ .

Graph:

To draw the graph of parametric equations $x=\left(2\cos t\right)+h$ and $y=\left(2\sin t\right)+k$ by the graphing calculator, follow the steps given below.

Step 1: Press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key to graph the functions. Now, press $\boxed{\text{MODE}}$ key to select Func, and then Par under the Normal to draw parametric equations.

Step 2: For $h=0$ and $k=-2$ , enter the keystrokes $\boxed{2}\boxed{\text{COS}}\boxed{\text{T}}\boxed{)}$ after the symbol ${\text{X}}_{\text{1T}}$ for the equation $x=2\cos t$ . Now, press $\boxed{\text{ENTER}}$ and enter the keystrokes $\boxed{2}\boxed{\text{SIN}}\boxed{\text{T}}\boxed{)}\boxed{-}\boxed{2}$ after the symbol ${\text{Y}}_{\text{1T}}$ for the equation $y=2\sin t-2$ .

Similarly, enter the equations for different values of $k=-1$ , $k=1$ and $k=2$ .

Step 3: Press $\boxed{\text{WINDOW}}$ key and set the parametric interval at $\left[0,2\pi \right]$ . Also, set the window at $\left[-6,6\right]$ by $\left[-4,4\right]$ . Press $\boxed{\text{GRAPH}}$ key to produce the graph of function as shown below.

  

Figure (2)

Interpretation: From the graph it can be observed that graph is a circle of radius 2 with center $\left(0,k\right)$ . The graph shifts vertically as the values of $k$ changes.

(c)

To determine

To find: The parametrization for the circle with radius 5 and center $\left(2,-3\right)$ .

Answer to Problem 44E

The parametrization for the circle with radius 5 and center $\left(2,-3\right)$ are $x=\left(5\cos t\right)+2$ and $y=\left(5\sin t\right)-3$ .

Explanation of Solution

Given information:The given parametric equations are $x=\left(2\cos t\right)+h$ and $y=\left(2\sin t\right)+k$ . The radius is 5 and the center is $\left(2,-3\right)$ .

Calculation:

It is given that the center of the circle is $\left(2,-3\right)$ . So, compare it with the general form of center of a circle $\left(h,k\right)$ . So,

  $h=2$ and $k=-3$

Also, the radius of the circle is 5. So, the coefficient of $\cos t$ and $\sin t$ will be 5.

The parametrization for the circle with radius 5 are:

  $x=\left(5\cos t\right)+2$

And

  $y=\left(5\sin t\right)-3$

Therefore, the parametrization for the circle with radius 5 and center $\left(2,-3\right)$ are $x=\left(5\cos t\right)+2$ and $y=\left(5\sin t\right)-3$ .