Chapter 8.4, Problem 62E

(a)

To determine

To find: the parametric equations for the curve.

Answer to Problem 62E

The parametric equations are $x=a\cos \theta$ and $y=b\sin \theta$

Explanation of Solution

Given:

  

Calculation:

Consider the following sketch of a concentric circles curve. Create a diagram of all the triangles and length using simple trigonometry

  

Create $a$ the first equation for the point $P$ . In order to do this create a triangle as seen in the sketch and find the $x$ length to the point $P$ .

  $x=a\cos \theta$ Using simple trigonometry of right triangles

Create $a$ the first equation for the point $P$ . In order to do this create a triangle as seen in the sketch and find the $y$ length to the point $P$ .

  $y=b\sin \theta$ Using simple trigonometry of right triangles

Conclusion:

Hence, the parametric equations are $x=a\cos \theta$ and $y=b\sin \theta$

(b)

To determine

To draw: a Graph of the curve using a graphing device.

Answer to Problem 62E

Hence the resultant graph is:

  

Explanation of Solution

Given:

  $a=3$ and $b=2$

Calculation:

Use a graphing device to plot the curve for $a=3,b=2$

  

Conclusion:

Hence the resultant graph is drawn.

(c)

To determine

To eliminate: the parameter, and identify the curve.

Answer to Problem 62E

The parameter is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

The curve is an ellipse.

Explanation of Solution

Calculation:

Now eliminate the parameter

  $\frac{x}{a}=\cos \theta$ isolate the trigonometric functions

  $\frac{y}{b}=\sin \theta$

  $\frac{x^2}{a^2}={\cos }^2\theta$ Square both sides

  $\frac{y^2}{b^2}={\sin }^2\theta$

  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ Write an equation using ${\sin }^2\theta +{\cos }^2\theta =1$

The curve is an ellipse.

Conclusion:

Hence the parameter is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

The curve is an ellipse.