Chapter 0.4, Problem 44E

(a.)

To determine

The graphs of the given parametric equations in the parametric interval $\left[0,2\pi \right]$ , for $k=0$ and $h=-2,-1,1,2$ in turn; and the role of $h$ .

Answer to Problem 44E

The graphs of the given parametric equations in the parametric interval $\left[0,2\pi \right]$ , for $k=0$ and $h=-2,-1,1,2$ in turn have been provided; and it has been determined that $h$ is the $x$ -coordinate of the center of the circle formed by graphing the given parametric equations.

Explanation of Solution

Given:

The parametric equations; $x=2\cos t+h$ , $y=2\sin t+k$ .

Concept used:

The parametric equations are graphed for each of the given values of $h$ , $k$ .

Calculation:

The given parametric equations are $x=2\cos t+h$ and $y=2\sin t+k$ .

The graphs of these parametric equations in the parametric interval $\left[0,2\pi \right]$ are as follows:

When $k=0$ and $h=-2$:

  

When $k=0$ and $h=-1$:

  

When $k=0$ and $h=1$:

  

When $k=0$ and $h=2$:

  

It can be seen that $h$ is the $x$ -coordinate of the center of the circle formed by graphing the given parametric equations.

Conclusion:

The graphs of the given parametric equations in the parametric interval $\left[0,2\pi \right]$ , for $k=0$ and $h=-2,-1,1,2$ in turn have been provided; and it has been determined that $h$ is the $x$ -coordinate of the center of the circle formed by graphing the given parametric equations.

(b.)

To determine

The graphs of the given parametric equations in the parametric interval $\left[0,2\pi \right]$ , for $h=0$ and $k=-2,-1,1,2$ in turn; and the role of $k$ .

Answer to Problem 44E

The graphs of the given parametric equations in the parametric interval $\left[0,2\pi \right]$ , for $h=0$ and $k=-2,-1,1,2$ in turn have been provided; and it has been determined that $k$ is the $y$ -coordinate of the center of the circle formed by graphing the given parametric equations.

Explanation of Solution

Given:

The parametric equations; $x=2\cos t+h$ , $y=2\sin t+k$ .

Concept used:

The parametric equations are graphed for each of the given values of $h$ , $k$ .

Calculation:

The given parametric equations are $x=2\cos t+h$ and $y=2\sin t+k$ .

The graphs of these parametric equations in the parametric interval $\left[0,2\pi \right]$ are as follows:

When $h=0$ and $k=-2$:

  

When $h=0$ and $k=-1$:

  

When $h=0$ and $k=1$:

  

When $h=0$ and $k=2$:

  

It can be seen that $k$ is the $y$ -coordinate of the center of the circle formed by graphing the given parametric equations.

Conclusion:

The graphs of the given parametric equations in the parametric interval $\left[0,2\pi \right]$ , for $h=0$ and $k=-2,-1,1,2$ in turn have been provided; and it has been determined that $k$ is the $y$ -coordinate of the center of the circle formed by graphing the given parametric equations.

(c.)

To determine

A parametrization of the given circle.

Answer to Problem 44E

It has been determined that the parametrization of the given circle is $x=5\cos t+2$ , $y=5\sin t-3$ in the parametric interval $\left[0,2\pi \right]$ .

Explanation of Solution

Given:

The circle with radius $5$ and center at $\left(2,-3\right)$ .

Concept used:

The parametrization; $x=a\cos t+h$ , $y=a\sin t+k$ in the parametric interval $\left[0,2\pi \right]$ ; gives a circle of radius $a$ whose center is at $\left(h,k\right)$ .

Calculation:

As seen in the previous parts, the parametrization; $x=a\cos t+h$ , $y=a\sin t+k$ in the parametric interval $\left[0,2\pi \right]$ ; gives a circle of radius $a$ whose center is at $\left(h,k\right)$ .

The given circle has radius $5$ .

Then, here $a=5$ .

The center of the given circle is at $\left(2,-3\right)$ .

Then, here $\left(h,k\right)=\left(2,-3\right)$ .

Put these values in $x=a\cos t+h$ , $y=a\sin t+k$ to get,

  $x=5\cos t+2$ ,

  $y=5\sin t-3$

So, the parametrization of the given circle is $x=5\cos t+2$ , $y=5\sin t-3$ in the parametric interval $\left[0,2\pi \right]$ .

Conclusion:

It has been determined that the parametrization of the given circle is $x=5\cos t+2$ , $y=5\sin t-3$ in the parametric interval $\left[0,2\pi \right]$ .

(d.)

To determine

A parameterization for the given ellipse.

Answer to Problem 44E

It has been determined that the parameterization for the given ellipse is $x=5\cos t-3$ , $y=2\sin t+4$ in the parametric interval $\left[0,2\pi \right]$ .

Explanation of Solution

Given:

An ellipse centered at $\left(-3,4\right)$ with semi-major axis of length $5$ parallel to $x$ -axis and semi-minor axis of length $2$ parallel to $y$ -axis.

Concept used:

The parametrization of an ellipse centered at $\left(h,k\right)$ with semi-major axis of length $a$ parallel to $x$ -axis and semi-minor axis of length $b$ parallel to $y$ -axis; is given as $x=a\cos t+h$ , $y=b\sin t+k$ , in the parametric interval $\left[0,2\pi \right]$ .

Calculation:

The given ellipse is centered at $\left(-3,4\right)$ .

Then, here $\left(h,k\right)=\left(-3,4\right)$ .

The length of the semi-major axis of the given ellipse, parallel to the $x$ -axis, is $5$ .

Then, here $a=5$ .

The length of the semi-minor axis of the given ellipse, parallel to the $y$ -axis, is $2$ .

Then, here $b=2$ .

Put these values in $x=a\cos t+h$ , $y=b\sin t+k$ to get,

  $x=5\cos t-3$ ,

  $y=2\sin t+4$

So, the required parametrization is $x=5\cos t-3$ , $y=2\sin t+4$ in the parametric interval $\left[0,2\pi \right]$ .

Conclusion:

It has been determined that the parameterization for the given ellipse is $x=5\cos t-3$ , $y=2\sin t+4$ in the parametric interval $\left[0,2\pi \right]$ .