Chapter 10

Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y.\) $$x=\sin 8 t, y=2 \cos 8 t$$

Consider an ellipse to be the set of points in a plane whose distances from two fixed points have a constant sum 2 \(a .\) Derive the equation of an ellipse. Assume the two fixed points are on the \(x\) -axis equidistant from the origin.

Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y.\) $$x=t, y=\sqrt{4-t^{2}}$$

Consider a hyperbola to be the set of points in a plane whose distances from two fixed points have a constant difference of \(2 a\) or \(-2 a\). Derive the equation of a hyperbola. Assume the two fixed points are on the \(x\) -axis equidistant from the origin.

Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y.\) $$x=\sqrt{t+1}, y=\frac{1}{t+1}$$

Show that the set of points equidistant from a circle and a line not passing through the circle is a parabola. Assume the circle, line, and parabola lie in the same plane.

Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y.\) $$x=\tan t, y=\sec ^{2} t-1$$

Show that the polar equation of an ellipse or a hyperbola with one focus at the origin, major axis of length \(2 a\) on the \(x\) -axis, and eccentricity \(e\) is $$r=\frac{a\left(1-e^{2}\right)}{1+e \cos \theta}$$

Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y.\) \(x=a \sin ^{n} t, y=b \cos ^{n} t,\) where \(a\) and \(b\) are real numbers and \(n\) is a positive integer

Suppose that two hyperbolas with eccentricities \(e\) and \(E\) have perpendicular major axes and share a set of asymptotes. Show that \(e^{-2}+E^{-2}=1\)

Slopes of tangent lines Find all the points at which the following curves have the given slope. $$x=4 \cos t, y=4 \sin t ; \text { slope }=\frac{1}{2}$$

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The lines tangent to the endpoints of any focal chord of a parabola \(y^{2}=4 p x\) intersect on the directrix and are perpendicular.

Slopes of tangent lines Find all the points at which the following curves have the given slope. $$x=2 \cos t, y=8 \sin t ; \text { slope }=-1$$

Consider the family of limaçons \(r=1+b \cos \theta .\) Describe how the curves change as \(b \rightarrow \infty\).

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. Let \(L\) be the latus rectum of the parabola \(y^{2}=4 p x,\) for \(p>0\) Let \(F\) be the focus of the parabola, \(P\) be any point on the parabola to the left of \(L,\) and \(D\) be the (shortest) distance between \(P\) and \(L\) Show that for all \(P, D+|F P|\) is a constant. Find the constant.

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The length of the latus rectum of the parabola \(y^{2}=4 p x\) or \(x^{2}=4 p y\) is \(4|p|\)

Find all the points at which the following curves have the given slope. $$x=2+\sqrt{t}, y=2-4 t ; \text { slope }=-8$$

Equations of the form \(r^{2}=a \sin 2 \theta\) and \(r^{2}=a \cos 2 \theta\) describe lemniscates (see Example 7 ). Graph the following lemniscates. \(r^{2}=4 \sin 2 \theta\)

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The length of the latus rectum of an ellipse centered at the origin is \(2 b^{2} / a=2 b \sqrt{1-e^{2}}\)

Find real numbers a and b such that equations \(A\) and \(B\) describe the same curve. \(A: x=10 \sin t, y=10 \cos t ; 0 \leq t \leq 2 \pi\) \(B: x=10 \sin 3 t, y=10 \cos 3 t ; a \leq t \leq b\)

Equations of the form \(r^{2}=a \sin 2 \theta\) and \(r^{2}=a \cos 2 \theta\) describe lemniscates (see Example 7 ). Graph the following lemniscates. \(r^{2}=-2 \sin 2 \theta\)

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The length of the latus rectum of a hyperbola centered at the origin is \(2 b^{2} / a=2 b \sqrt{e^{2}-1}\)

Find real numbers a and b such that equations \(A\) and \(B\) describe the same curve. \(A: x=t+t^{3}, y=3+t^{2} ;-2 \leq t \leq 2\) \(B: x=t^{1 / 3}+t, y=3+t^{2 / 3} ; a \leq t \leq b\)

Equations of the form \(r^{2}=a \sin 2 \theta\) and \(r^{2}=a \cos 2 \theta\) describe lemniscates (see Example 7 ). Graph the following lemniscates. \(r^{2}=-8 \cos 2 \theta\)

Show that an ellipse and a hyperbola that have the same two foci intersect at right angles.

Consider the following Lissajous curves. Graph the curve and estimate the coordinates of the points on the curve at which there is (a) a horizontal tangent line and (b) a vertical tangent line. (See the Guided Project Parametric art for more on Lissajous curves.) $$\begin{aligned}&x=\sin 2 t, y=2 \sin t\\\&0 \leq t \leq 2 \pi\end{aligned}$$

Equations of the form \(r=a \sin m \theta\) or \(r=a \cos m \theta,\) where \(a\) is a real number and \(m\) is a positive integer, have graphs known as roses (see Example 6 ). Graph the following roses. \(r=\sin 2 \theta\)

Show that the vertical distance between a hyperbola \(x^{2} / a^{2}-y^{2} / b^{2}=1\) and its asymptote \(y=b x / a\) approaches zero as \(x \rightarrow \infty,\) where \(0

Consider the following Lissajous curves. Graph the curve and estimate the coordinates of the points on the curve at which there is (a) a horizontal tangent line and (b) a vertical tangent line. (See the Guided Project Parametric art for more on Lissajous curves.) $$\begin{aligned}&x=\sin 4 t, y=\sin 3 t\\\&0 \leq t \leq 2 \pi\end{aligned}$$

Equations of the form \(r=a \sin m \theta\) or \(r=a \cos m \theta,\) where \(a\) is a real number and \(m\) is a positive integer, have graphs known as roses (see Example 6 ). Graph the following roses. \(r=4 \cos 3 \theta\)

Sector of a hyperbola Let \(H\) be the right branch of the hyperbola \(x^{2}-y^{2}=1\) and let \(\ell\) be the line \(y=m(x-2)\) that passes through the point (2,0) with slope \(m,\) where \(-\infty < m < \infty\). Let \(R\) be the region in the first quadrant bounded by \(H\) and \(\ell\) (see figure). Let \(A(m)\) be the area of \(R .\) Note that for some values of \(m\) \(A(m)\) is not defined. a. Find the \(x\) -coordinates of the intersection points between \(H\) and \(\ell\) as functions of \(m ;\) call them \(u(m)\) and \(v(m),\) where \(v(m) > u(m) > 1 .\) For what values of \(m\) are there two intersection points? b. Evaluate \(\lim _{m \rightarrow 1^{+}} u(m)\) and \(\lim _{m \rightarrow 1^{+}} v(m)\) c. Evaluate \(\lim _{m \rightarrow \infty} u(m)\) and \(\lim _{m \rightarrow \infty} v(m)\) d. Evaluate and interpret \(\lim _{m \rightarrow \infty} A(m)\)

The Lamé curve described by \(\left|\frac{x}{a}\right|^{n}+\left|\frac{y}{b}\right|^{n}=1,\) where \(a, b,\) and \(n\) are positive real numbers, is a generalization of an ellipse. a. Express this equation in parametric form (four pairs of equations are needed). b. Graph the curve for \(a=4\) and \(b=2,\) for various values of \(n\) c. Describe how the curves change as \(n\) increases.

Equations of the form \(r=a \sin m \theta\) or \(r=a \cos m \theta,\) where \(a\) is a real number and \(m\) is a positive integer, have graphs known as roses (see Example 6 ). Graph the following roses. \(r=2 \sin 4 \theta\)

A family of curves called hyperbolas (discussed in Section 10.4 ) has the parametric equations \(x=a\) tan \(t\) \(y=b \sec t,\) for \(-\pi

Consider the parametric equations $$ x=a \cos t+b \sin t, \quad y=c \cos t+d \sin t $$ where \(a, b, c,\) and \(d\) are real numbers. a. Show that (apart from a set of special cases) the equations describe an ellipse of the form \(A x^{2}+B x y+C y^{2}=K,\) where \(A, B, C,\) and \(K\) are constants. b. Show that (apart from a set of special cases), the equations describe an ellipse with its axes aligned with the \(x\) - and \(y\) -axes provided \(a b+c d=0\) c. Show that the equations describe a circle provided \(a b+c d=0\) and \(c^{2}+d^{2}=a^{2}+b^{2} \neq 0\)

A trochoid is the path followed by a point \(b\) units from the center of a wheel of radius \(a\) as the wheel rolls along the \(x\) -axis. Its parametric description is \(x=a t-b \sin t, y=a-b \cos t .\) Choose specific values of \(a\) and \(b,\) and use a graphing utility to plot different trochoids. In particular, explore the difference between the cases \(a>b\) and \(a

Show that the graph of \(r=a \sin m \theta\) or \(r=a \cos m \theta\) is a rose with \(m\) leaves if \(m\) is an odd integer and a rose with \(2 m\) leaves if \(m\) is an even integer.

An epitrochoid is the path of a point on a circle of radius \(b\) as it rolls on the outside of a circle of radius \(a\). It is described by the equations $$\begin{array}{l}x=(a+b) \cos t-c \cos \left(\frac{(a+b) t}{b}\right) \\\y=(a+b) \sin t-c \sin \left(\frac{(a+b) t}{b}\right)\end{array}$$ Use a graphing utility to explore the dependence of the curve on the parameters \(a, b,\) and \(c.\)

A general hypocycloid is described by the equations $$\begin{aligned}&x=(a-b) \cos t+b \cos \left(\frac{(a-b) t}{b}\right)\\\&y=(a-b) \sin t-b \sin \left(\frac{(a-b) t}{b}\right)\end{aligned}$$ Use a graphing utility to explore the dependence of the curve on the parameters \(a\) and \(b\)

Graph the following spirals. Indicate the direction in which the spiral is generated as \(\theta\) increases, where \(\theta>0 .\) Let \(a=1\) and \(a=-1\). Logarithmic spiral: \(r=e^{a \theta}\)

An idealized model of the path of a moon (relative to the Sun) moving with constant speed in a circular orbit around a planet, where the planet in turn revolves around the Sun, is given by the parametric equations $$x(\theta)=a \cos \theta+\cos n \theta, y(\theta)=a \sin \theta+\sin n \theta.$$ The distance from the moon to the planet is taken to be \(1,\) the distance from the planet to the Sun is \(a,\) and \(n\) is the number of times the moon orbits the planet for every 1 revolution of the planet around the Sun. Plot the graph of the path of a moon for the given constants; then conjecture which values of \(n\) produce loops for a fixed value of \(a\) a. \(a=4, n=3\) b. \(a=4, n=4 \) c. \(a=4, n=5\)

Graph the following spirals. Indicate the direction in which the spiral is generated as \(\theta\) increases, where \(\theta>0 .\) Let \(a=1\) and \(a=-1\). Hyperbolic spiral: \(r=a / \theta\)

Points at which the graphs of \(r=f(\theta)\) and \(r=g(\theta)\) intersect must be determined carefully. Solving \(f(\theta)=g(\theta)\) identifies some-but perhaps not all-intersection points. The reason is that the curves may pass through the same point for different values of \(\theta .\) Use analytical methods and a graphing utility to find all the intersection points of the following curves. \(r=2 \cos \theta\) and \(r=1+\cos \theta\)

A plane traveling horizontally at \(80 \mathrm{m} / \mathrm{s}\) over flat ground at an elevation of 3000 m releases an emergency packet. The trajectory of the packet is given by $$x=80 t, \quad y=-4.9 t^{2}+3000, \quad \text { for } t \geq 0$$ where the origin is the point on the ground directly beneath the plane at the moment of the release. Graph the trajectory of the packet and find the coordinates of the point where the packet lands.

A plane traveling horizontally at \(100 \mathrm{m} / \mathrm{s}\) over flat ground at an elevation of \(4000 \mathrm{m}\) must drop an emergency packet on a target on the ground. The trajectory of the packet is given by $$x=100 t, \quad y=-4.9 t^{2}+4000, \quad \text { for } t \geq 0,$$ where the origin is the point on the ground directly beneath the plane at the moment of the release. How many horizontal meters before the target should the packet be released in order to hit the target?

Points at which the graphs of \(r=f(\theta)\) and \(r=g(\theta)\) intersect must be determined carefully. Solving \(f(\theta)=g(\theta)\) identifies some-but perhaps not all-intersection points. The reason is that the curves may pass through the same point for different values of \(\theta .\) Use analytical methods and a graphing utility to find all the intersection points of the following curves. \(r=1-\sin \theta\) and \(r=1+\cos \theta\)

Explain and carry out a method for graphing the curve \(x=1+\cos ^{2} y-\sin ^{2} y\) using parametric equations and a graphing utility.

The butterfly curve of Example 8 is enhanced by adding a term: $$r=e^{\sin \theta}-2 \cos 4 \theta+\sin ^{5}(\theta / 12), \quad \text { for } 0 \leq \theta \leq 24 \pi$$ a. Graph the curve. b. Explain why the new term produces the observed effect. (Source: S. Wagon and E. Packel, Animating Calculus, Freeman, 1994)

Assume a curve is given by the parametric equations \(x=f(t)\) and \(y=g(t),\) where \(f\) and \(g\) are twice differentiable. Use the Chain Rule to show that $$y^{\prime \prime}(x)=\frac{f^{\prime}(t) g^{\prime \prime}(t)-g^{\prime}(t) f^{\prime \prime}(t)}{\left(f^{\prime}(t)\right)^{3}}.$$

Consider the curve \(r=f(\theta)=\cos a^{\theta}-1.5\) where \(a=(1+12 \pi)^{1 /(2 \pi)} \approx 1.78933\) (see figure). a. Show that \(f(0)=f(2 \pi)\) and find the point on the curve that corresponds to \(\theta=0\) and \(\theta=2 \pi\) b. Is the same curve produced over the intervals \([-\pi, \pi]\) and \([0,2 \pi] ?\) c. Let \(f(\theta)=\cos a^{\theta}-b,\) where \(a=(1+2 k \pi)^{1 /(2 \pi)}, k\) is an integer, and \(b\) is a real number. Show that \(f(0)=f(2 \pi)\) and that the curve closes on itself. d. Plot the curve with various values of \(k\). How many fingers can you produce?