Chapter 11

Find an equation for the hyperbola that satisfies the given conditions. Foci \(( \pm 5,0),\) length of transverse axis 6

Use a graphing device to draw the curve represented by the parametric equations. $$ x=2 \sin t, \quad y=\cos 4 t $$

Find an equation for the ellipse that satisfies the given conditions. Eccentricity \(\frac{1}{9},\) foci \((0, \pm 2)\)

Path of a Cannonball A cannon fires a cannonball as shown in the figure. The path of the cannonball is a parabola with vertex at the highest point of the path. If the cannon ball lands 1600 ft from the cannon and the highest point it reaches is 3200 ft above the ground, find an equation for the path of the cannonball. Place the origin at the location of the cannon.

Find an equation for the hyperbola that satisfies the given conditions. Foci \((0, \pm 1),\) length of transverse axis 1

Use a graphing device to draw the curve represented by the parametric equations. $$ x=3 \sin 5 t, \quad y=5 \cos 3 t $$

Find an equation for the ellipse that satisfies the given conditions. Eccentricity \(0.8,\) foci \(( \pm 1.5,0)\)

Orbit of a Satellite A satellite is in an elliptical orbit round the earth with the center of the earth at one focus. The height of the satellite above the earth varies between 140 \(\mathrm{mi}\) and 440 \(\mathrm{mi}\) . Assume the earth is a sphere with radius 3960 \(\mathrm{mi}\) . Find an equation for the path of the satellite with the origin at the center of the earth.

(a) Show that the asymptotes of the hyperbola \(x^{2}-y^{2}=5\) are perpendicular to each other. (b) Find an equation for the hyperbola with foci \(( \pm c, 0)\) and with asymptotes perpendicular to each other.

Use a graphing device to draw the curve represented by the parametric equations. $$ x=\sin 4 t, \quad y=\cos 3 t $$

Find an equation for the ellipse that satisfies the given conditions. Eccentricity \(\sqrt{3} / 2,\) foci on \(y\) -axis, length of major axis 4

The hyperbolas $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad \text { and } \quad \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1$$ are said to be conjugate to each other. (a) Show that the hyperbolas $$x^{2}-4 y^{2}+16=0 \quad \text { and } \quad 4 y^{2}-x^{2}+16=0$$ are conjugate to each other, and sketch their graphs on the same coordinate axes. (b) What do the hyperbolas of part (a) have in common? (c) Show that any pair of conjugate hyperbolas have the relationship you discovered in part (b).

Use a graphing device to draw the curve represented by the parametric equations. $$ x=\sin (\cos t), \quad y=\cos \left(t^{3 / 2}\right), \quad 0 \leq t \leq 2 \pi $$

Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes and label the points of intersection. $$ \left\\{\begin{array}{l}{4 x^{2}+y^{2}=4} \\ {4 x^{2}+9 y^{2}=36}\end{array}\right. $$

In the derivation of the equation of the hyperbola at the beginning of this section, we said that the equation $$\sqrt{(x+c)^{2}+y^{2}}-\sqrt{(x-c)^{2}+y^{2}}=\pm 2 a$$ simplifies to $$\left(c^{2}-a^{2}\right) x^{2}-a^{2} y^{2}=a^{2}\left(c^{2}-a^{2}\right)$$ Supply the steps needed to show this.

Use a graphing device to draw the curve represented by the parametric equations. $$ x=2 \cos t+\cos 2 t, \quad y=2 \sin t-\sin 2 t $$

Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes and label the points of intersection. $$ \left\\{\begin{array}{l}{\frac{x^{2}}{16}+\frac{y^{2}}{9}=1} \\\ {\frac{x^{2}}{9}+\frac{y^{2}}{16}=1}\end{array}\right. $$

(a) For the hyperbola $$\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$$ determine the values of \(a, b,\) and \(c,\) and find the coordinates of the foci \(F_{1}\) and \(F_{2} .\) (b) Show that the point \(P\left(5, \frac{16}{3}\right)\) lies on this hyperbola. (c) Find \(d\left(P, F_{1}\right)\) and \(d\left(P, F_{2}\right)\) . (d) Verify that the difference between \(d\left(P, F_{1}\right)\) and \(d\left(P, F_{2}\right)\) is 2\(a .\)

A polar equation is given. (a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$ r=2^{\theta / 12}, \quad 0 \leq \theta \leq 4 \pi $$

Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes and label the points of intersection. $$ \left\\{\begin{array}{c}{100 x^{2}+25 y^{2}=100} \\\ {x^{2}+\frac{y^{2}}{9}=1}\end{array}\right. $$

Hyperbolas are called confocal if they have the same foci. (a) Show that the hyperbolas $$\frac{y^{2}}{k}-\frac{x^{2}}{16-k}=1 \quad \text { with } 0< k< 16$$ are confocal. (b) Use a graphing device to draw the top branches of the family of hyperbolas in part (a) for \(k=1,4,8,\) and \(12 .\) How does the shape of the graph change as \(k\) increases?

A polar equation is given. (a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$ r=\sin \theta+2 \cos \theta $$

(a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$ r=\frac{4}{2-\cos \theta} $$

(a) Use a graphing device to sketch the top half (the portion in the first and second quadrants) of the family of ellipses \(x^{2}+k y^{2}=100\) for \(k=4,10,25,\) and \(50 .\) (b) What do the members of this family of ellipses have in common? How do they differ?

A polar equation is given. (a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$ r=2^{\sin \theta} $$

If \(k>0\) , the following equation represents an ellipse: $$ \frac{x^{2}}{k}+\frac{y^{2}}{4+k}=1 $$ Show that all the ellipses represented by this equation have the same foci, no matter what the value of \(k\) .

Two stones are dropped simultaneously in a calm pool of water. The crests of the resulting waves form equally spaced concentric circles, as shown in the figures. The waves interact with each other to create certain interference patterns. (a) Explain why the red dots lie on an ellipse. (b) Explain why the blue dots lie on a hyperbola.

Several examples of the uses of hyperbolas are given in the text. Find other situations in real life where hyperbolas occur. Consult a scientific encyclopedia in the reference section of your library, or search the Internet.

(a) Find equations for the family of parabolas with vertex at the origin and with directrixes \(y=\frac{1}{2}, y=1, y=4\) and \(y=8\). (b) Draw the graphs. What do you conclude?

(a) Find equations for the family of parabolas with vertex at the origin, focus on the positive \(y\)-axis, and with focal diameters 1, 2, 4, and 8. (b) Draw the graphs. What do you conclude?

Lunar Orbit For an object in an elliptical orbit around the moon, the points in the orbit that are closest to and farthest from the center of the moon are called perilune and apolune, respectively. These are the vertices of the orbit. The center of the moon is at one focus of the orbit. The Apollo 11 spacecraft was placed in a lunar orbit with perilune at 68 \(\mathrm{mi}\) and apolune at 195 \(\mathrm{mi}\) above the surface of the moon. Assuming the moon is a sphere of radius 1075 \(\mathrm{mi}\) , find an equation for the orbit of Apollo \(11 .\) (Place the coordinate axes so that the origin is at the center of the orbit and the foci are located on the \(x\) -axis.)

A lamp with a parabolic reflector is shown in the figure. The bulb is placed at the focus and the focal diameter is 12 cm. (a) Find an equation of the parabola. (b) Find the diameter \(d(C, D)\) of the opening, 20 cm from the vertex.

Plywood Ellipse A carpenter wishes to construct an elliptical table top from a sheet of plywood, 4 \(\mathrm{ft}\) by 8 \(\mathrm{ft}\) . He will trace out the ellipse using the "thumbtack and string" method illustrated in Figures 2 and \(3 .\) What length of string should he use, and how far apart should the tacks be located, if the ellipse is to be the largest possible that can be cut out of the plywood sheet?

A reflector for a satellite dish is parabolic in cross section, with the receiver at the focus \(F\). The reflector is 1 ft deep and 20 ft wide from rim to rim (see the figure). How far is the receiver from the vertex of the parabolic reflector?

Sunburst Window \(A\) "sunburst" window above a door- way is constructed in the shape of the top half of an ellipse, as shown in the figure. The window is 20 in. tall at its high- est point and 80 in. wide at the bottom. Find the height of the window 25 in. from the center of the base.

In a suspension bridge the shape of the suspension cables is parabolic. The bridge shown in the figure has towers that are 600 m apart, and the lowest point of the suspension cables is 150 m below the top of the towers. Find the equation of the parabolic part of the cables, placing the origin of the coordinate system at the vertex. NOTE This equation is used to find the length of cable needed in the construction of the bridge.

Drawing an Ellipse on a Blackboard Try drawing an ellipse as accurately as possible on a blackboard. How would a piece of string and two friends help this process?

The Hale telescope at the Mount Palomar Observatory has a 200-in. mirror, as shown. The mirror is constructed in a parabolic shape that collects light from the stars and focuses it at the prime focus, that is, the focus of the parabola. The mirror is 3.79 in. deep at its center. Find the focal length of this parabolic mirror, that is, the distance from the vertex to the focus.

Several examples of the uses of parabolas are given in the text. Find other situations in real life where parabolas occur. Consult a scientific encyclopedia in the reference section of your library, or search the Internet.

How Wide Is an Ellipse at Its Foci? A latus rectum for an ellipse is a line segment perpendicular to the major axis at a focus, with endpoints on the ellipse, as shown. Show that the length of a latus rectum is 2\(b^{2} / a\) for the ellipse $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad \text { with } a>b $$

Is It an Ellipse? A piece of paper is wrapped around a cylindrical bottle, and then a compass is used to draw a circle on the paper, as shown in the figure. When the paper is laid flat, is the shape drawn on the paper an ellipse? (You don't need to prove your answer, but you may want to do the experiment and see what you get.)

Eliminate the parameter \(\theta\) in the parametric equations for the cycloid (Example 6 ) to obtain a rectangular coordinate equation for the section of the curve given by \(0 \leq \theta \leq \pi\) .

Spiral Path of a Dog \(\mathrm{A}\) dog is tied to a circular tree trunk of radius 1 \(\mathrm{ft}\) by a long leash. He has managed to wrap the entire leash around the tree while playing in the yard, and finds himself at the point \((1,0)\) in the figure. Seeing a squirrel, he runs around the tree counterclockwise, keeping the leash taut while chasing the intruder. (a) Show that parametric equations for the dog's path (called an involute of a circle) are \(x=\cos \theta+\theta \sin \theta \quad y=\sin \theta-\theta \cos \theta\) \([\text {Hint: Note that the leash is always tangent to the tree, }\) \(\text { so } O T \text { is perpendicular to } T D .]\) (b) Graph the path of the dog for \(0 \leq \theta \leq 4 \pi\)

Different Ways of Tracing Out a Curve The curves \(C,\) \(D, E,\) and \(F\) are defined parametrically as follows, where the parameter \(t\) takes on all real values unless otherwise stated: \(C : \quad x=t, \quad y=t^{2}\) \(D : \quad x=\sqrt{t}, \quad y=t, \quad t \geq 0\) \(E : \quad x=\sin t, \quad y=\sin ^{2} t\) \(F : \quad x=3^{t}, \quad y=3^{2 t}\) (a) Show that the points on all four of these curves satisfy the same rectangular coordinate equation. (b) Draw the graph of each curve and explain how the curves differ from one another.