Chapter 12, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

1. a. A set of parametric equations for a given curve is always unique.
2. b. The equations x = e^t, y = 2e^t, for − ∞ < t < ∞, describe a line passing through the origin with slope 2.
3. c. The polar coordinates (3, −3π/4) and (−3, π/4) describe the same point in the plane.
4. d. The area of the region between the inner and outer loops of the limaçon r = f(θ) = 1 − 4 cos θ is $\frac{1}{2}{\displaystyle {\int }_0^{2\pi }f{(\theta )}^{2}d\theta }$.
5. e. The hyperbola y²/2 − x²/4 = 1 has no x-intercept.
6. f. The equation x² + 4y² − 2x = 3 describes an ellipse.

To determine

Whether the statement, “A set of parametric equations for a given curve is always unique” is true or false.

Answer to Problem 1RE

The given statement is false.

Explanation of Solution

Consider the parametric equation of a circle as $x=r\cos t,y=r\sin t;0\le t\le 2\pi$.

Substitute the parametric equations to the equation of circle as follows.

$\begin{array}{c}{x}^2+y^2=r^2{\cos }^{2}t+r^2{\sin }^{2}t\\ =r^2\left({\cos }^{2}t+{\sin }^{2}t\right)\\ =r^2\end{array}$

Also, note that the other parametric equation of the circle is $x=r\sin t,y=r\cos t;0\le t\le 2\pi$.

$\begin{array}{c}{x}^2+y^2=r^2{\sin }^{2}t+r^2{\cos }^{2}t\\ =r^2\left({\sin }^{2}t+{\cos }^{2}t\right)\\ =r^2\end{array}$

Therefore, note that equations of the curve are same but the parametric equations are different.

Therefore, the given statement is false.

To determine

Whether the statement, “The equations $x=e^t,y=2e^t$ for $-\infty <t<\infty$, describe a line passing through the origin with slope 2” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Note that the given equation is $x=e^t,y=2e^t;-\infty <t<\infty$.

For any value of t the value of $e^t>0$ thus, $x,y>0$ for any value of t.

Thus, the given equation will represent a part of the line with slope 2 in first quadrant.

Therefore, the given statement is true.

To determine

Whether the statement, “the polar coordinates $\left(3,-\frac{3\pi }{4}\right)\text{ and }\left(-3,\frac{\pi }{4}\right)$ describe the same point in the plane”, is true or false.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

The given parametric equation is $\left(3,-\frac{3\pi }{4}\right)\text{ and }\left(-3,\frac{\pi }{4}\right)$.

The Cartesian coordinate $\left(x,y\right)$ corresponding to polar coordinates $\left(r,\theta \right)$ are given by $\left(r\cos \theta ,r\sin \theta \right)$.

Therefore, the Cartesian coordinates corresponding to the point $\left(3,-\frac{3\pi }{4}\right)$ is given as follows.

$\begin{array}{c}\left(r\cos \theta ,r\sin \theta \right)=\left(3\cos \left(-\frac{3\pi }{4}\right),3\sin \left(-\frac{3\pi }{4}\right)\right)\\ =\left(3\left(-\frac{1}{\sqrt{2}}\right),3\left(-\frac{1}{\sqrt{2}}\right)\right)\\ =\left(-\frac{3}{\sqrt{2}},-\frac{3}{\sqrt{2}}\right)\end{array}$

Hence, the point $\left(3,-\frac{3\pi }{4}\right)$ represents the point $\left(-\frac{3}{\sqrt{2}},-\frac{3}{\sqrt{2}}\right)$.

Similarly, the Cartesian coordinates corresponding to the point $\left(-3,\frac{\pi }{4}\right)$ is given as follows.

$\begin{array}{c}\left(r\cos \theta ,r\sin \theta \right)=\left(-3\cos \left(\frac{\pi }{4}\right),-3\sin \left(\frac{\pi }{4}\right)\right)\\ =\left(-3\left(\frac{1}{\sqrt{2}}\right),-3\left(\frac{1}{\sqrt{2}}\right)\right)\\ =\left(-\frac{3}{\sqrt{2}},-\frac{3}{\sqrt{2}}\right)\end{array}$

Hence the point $\left(3,-\frac{3\pi }{4}\right)$ represents the point $\left(-\frac{3}{\sqrt{2}},-\frac{3}{\sqrt{2}}\right)$.

Therefore, the polar points $\left(3,-\frac{3\pi }{4}\right)\text{ and }\left(-3,\frac{\pi }{4}\right)$ represent the same point in the plane.

Therefore, the statement is false.

To determine

Whether the statement, “the area of the region between the inner and outer loops of the limacon $r=f\left(\theta \right)=1-4\cos \theta$ is $\frac{1}{2}{\displaystyle {\int }_0^{2\pi }f{\left(\theta \right)}^{2}d\theta }$” is true or false.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

The area inside the region described by a curve $r=f\left(\theta \right)$ is given by $A=\frac{1}{2}{\displaystyle {\int }_0^{2\pi }f{\left(\theta \right)}^{2}d\theta }$.

Therefore, the given integral $\frac{1}{2}{\displaystyle {\int }_0^{2\pi }f{\left(\theta \right)}^{2}d\theta }$ represents the area inside the inner loop of the limacon $r=f\left(\theta \right)=1-4\cos \theta$ and not the area between two loops.

Therefore, the statement is false.

To determine

Whether the statement, “the hyperbola $\frac{y^2}{2}-\frac{x^2}{4}=1$ has no x-intercept” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

To find the x-intercept substitute $y=0$ in $\frac{y^2}{2}-\frac{x^2}{4}=1$.

$\begin{array}{l}\frac{0^2}{2}-\frac{x^2}{4}=1\\ -x^2=4\\ x^2=-4\end{array}$

Note that the equation has no solution.

Thus, there does not exist a x-intercept for the hyperbola $\frac{y^2}{2}-\frac{x^2}{4}=1$.

Therefore, the given statement is true.

To determine

Whether the statement, “the equation $x^2+4y^2-2x=3$ describe an ellipse” is true or false.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Rewrite the given equation as $x^2-2x+1+4y^2=4$.

Rearrange the terms of the equation as follows.

$\begin{array}{l}{x}^2-2x+1+4y^2=4\\ {\left(x-1\right)}^2+4y^2=4\\ \frac{{\left(x-1\right)}^2}{4}+y^2=1\end{array}$

Which is the general equation of an ellipse with center at $\left(1,0\right)$.

Therefore, the given statement is true.