Chapter 10

In 1957, Russia launched Sputnik I. Its elliptical orbit around the earth reached maximum and minimum distances from the earth of 583 miles and 132 miles, respectively. Assuming that the center of the earth is one focus and that the earth is a sphere of radius 4000 miles, find the eccentricity of the orbit.

In many cases, polar graphs are related to each other by rotation. We explore that concept here. (a) How are the graphs of \(r=1+\sin (\theta-\pi / 3)\) and \(r=\) \(1+\sin (\theta+\pi / 3)\) related to the graph of \(r=1+\sin \theta ?\) (b) How is the graph of \(r=1+\sin \theta\) related to the graph of \(r=1-\sin \theta ?\) (c) How is the graph of \(r=1+\sin \theta\) related to the graph of \(r=1+\cos \theta ?\) (d) How is the graph of \(r=f(\theta)\) related to the graph of \(r=f(\theta-\alpha) ?\)

The orbit of the planet Pluto has an eccentricity \(0.249 .\) The closest that Pluto comes to the sun is \(29.65 \mathrm{AU}\), and the farthest is \(49.31\) AU. Find the major and minor diameters.

Investigate the family of curves given by \(r=a+b \cos (n(\theta+\phi))\) where \(a, b\), and \(\phi\) are real numbers and\(n\) is a positive integer. As you answer the following questions, be sure that you graph a sufficient number of examples to justify your conclusions. (a) How are the graphs for \(\phi=0\) related to those for which \(\phi \neq 0 ?\) (b) How does the graph change as \(n\) increases? (c) How do the relative magnitude and sign of \(a\) and \(b\) change the nature of the graph?

Find the area of the surface generated by revolving the curve \(x=2+\cos t, y=1+\sin t\), for \(0 \leq t \leq 2 \pi\) about the \(x\) -axis.

If two tangent lines to the ellipse \(9 x^{2}+4 y^{2}=36\) intersect the \(y\) -axis at \((0,6)\), find the points of tangency.

Investigate the family of curves defined by the polar equations \(r=|\cos n \theta|\), where \(n\) is some positive integer. How do the number of leaves depend on \(n\) ?

Find the area of the surface generated by revolving the curve \(x=(2 / 3) t^{3 / 2}, y=2 \sqrt{t}\), for \(0 \leq t \leq 2 \sqrt{3}\) about the \(y\) -axis.

If the tangent lines to the hyperbola \(9 x^{2}-y^{2}=36\) intersect the \(y\) -axis at \((0,6)\), find the points of tangency.

A curve \(C\) goes through the three points, \((-1,2),(0,0)\), and \((3,6)\). Find an equation for \(C\) if \(C\) is (a) a vertical parabola; (b) a horizontal parabola; (c) a circle.

Find the area of the surface generated by revolving the curve \(x=t+\sqrt{7}, y=t^{2} / 2+\sqrt{7} t\), for \(-\sqrt{7} \leq t \leq \sqrt{7}\) about the \(y\) -axis.

The slope of the tangent line to the hyperbola $$ 2 x^{2}-7 y^{2}-35=0 $$ at two points on the hyperbola is \(-\frac{2}{3}\). What are the coordinates of the points of tangency?

The ends of an elastic string with a knot at \(K(x, y)\) are attached to a fixed point \(A(a, b)\) and a point \(P\) on the rim of a wheel of radius \(r\) centered at \((0,0) .\) As the wheel turns, \(K\) traces a curve \(C\). Find the equation for \(C\). Assume that the string stays taut and stretches uniformly (i.e., \(\alpha=|K P| /|A P|\) is constant).

Sketch the reciprocal spiral given by \(r=c / \theta .\) For \(c>0\), does it unwind in the clockwise direction?

Find the area of the surface generated by revolving the curve \(x=t^{2} / 2+a t, y=t+a\), for \(-\sqrt{a} \leq t \leq \sqrt{a}\) about the \(x\) -axis.

Find the equations of the tangent lines to the ellipse \(x^{2}+2 y^{2}-2=0\) that are parallel to the line $$ 3 x-3 \sqrt{2} y-7=0 $$

Name the conic \(y^{2}=L x+K x^{2}\) according to the value of \(K\) and then show that in every case \(|L|\) is the length of the latus rectum of the conic. Assume that \(L \neq 0\).

Evaluate the integrals . $$ \int_{0}^{1}\left(x^{2}-4 y\right) d x, \text { where } x=t+1, y=t^{3}+4 $$

Find the area of the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\).

Show that the equations of the parabola and hyperbola with vertex \((a, 0)\) and focus \((c, 0), c>a>0\), can be written as \(y^{2}=4(c-a)(x-a)\) and \(y^{2}=\left(b^{2} / a^{2}\right)\left(x^{2}-a^{2}\right)\), respectively. Then use these expressions for \(y^{2}\) to show that the parabola is always "inside" the right branch of the hyperbola.

Evaluate the integrals $$ \int_{1}^{\sqrt{3}} x y d y, \text { where } x=\sec t, y=\tan t $$

Find the volume of the solid obtained by revolving the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\) about the \(y\) -axis.

The graph of \(x \cos \alpha+y \sin \alpha=d\) is a line. Show that the perpendicular distance from the origin to this line is \(|d|\) by making a rotation of axes through the angle \(\alpha .\)

. Find the area of the region between the curve \(x=e^{2 t}, y=e^{-t}\), and the \(x\) -axis from \(t=0\) to \(t=\ln 5 .\) Make a sketch.

The region bounded by the hyperbola $$ b^{2} x^{2}-a^{2} y^{2}=a^{2} b^{2} $$ and a vertical line through a focus is revolved about the \(x\) -axis. Find the volume of the resulting solid.

Transform the equation \(x^{1 / 2}+y^{1 / 2}=a^{1 / 2}\) by a rotation of axes through \(45^{\circ}\) and then square twice to eliminate radicals on variables. Identify the corresponding curve.

58\. The path of a projectile fired from level ground with a speed of \(v_{0}\) feet per second at an angle \(\alpha\) with the ground is given by the parametric equations $$ x=\left(v_{0} \cos \alpha\right) t, \quad y=-16 t^{2}+\left(v_{0} \sin \alpha\right) t $$ (a) Show that the path is a parabola. (b) Find the time of flight. (c) Show that the range (horizontal distance traveled) is \(\left(v_{0}^{2} / 32\right) \sin 2 \alpha\) (d) For a given \(v_{0}\), what value of \(\alpha\) gives the largest possible range?

Find the dimensions of the rectangle having the greatest possible area that can be inscribed in the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2} .\) Assume that the sides of the rectangle are parallel to the axes of the ellipse.

Find the points of \(x^{2}+14 x y+49 y^{2}=100\) that are closest to the origin.

. Let a circle of radius \(b\) roll, without slipping, inside a fixed circle of radius \(a, a>b .\) A point \(P\) on the rolling circle traces out a curve called a hypocycloid. Find parametric equations of the hypocycloid. Hint: Place the origin \(O\) of Cartesian coordinates at the center of the fixed, larger circle, and let the point \(A(a, 0)\) be one position of the tracing point \(P\). Denote by \(B\) the moving point of tangency of the two circles, and let \(t\), the radian measure of the angle \(A O B\), be the parameter (see Figure 11 ).

Find the point in the first quadrant where the two hyperbolas \(25 x^{2}-9 y^{2}=225\) and \(-25 x^{2}+18 y^{2}=450\) intersect.

Find the points of intersection of \(x^{2}+4 y^{2}=20\) and \(x+2 y=6\)

The curve traced by a point on a circle of radius \(b\) as it rolls without slipping on the outside of a fixed circle of radius \(a\) is called an epicycloid. Show that it has parametric equations $$ \begin{array}{l} x=(a+b) \cos t-b \cos \frac{a+b}{b} t \\ y=(a+b) \sin t-b \sin \frac{a+b}{b} t \end{array} $$

A ball placed at a focus of an elliptical billiard table is shot with tremendous force so that it continues to bounce off the cushions indefinitely. Describe its ultimate path? Hint: Draw a picture.

Show that, if \(A+C\) and \(\Delta=4 A C-B^{2}\) are both positive, then the graph of \(A x^{2}+B x y+C y^{2}=1\) is an ellipse (or circle) with area \(2 \pi / \sqrt{\Delta}\). (Recall from Problem 55 of Section \(10.2\) that the area of the ellipse \(x^{2} / p^{2}+y^{2} / q^{2}=1\) is \(\left.\pi p q .\right)\)

. Consider the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\). (a) Show that its perimeter is $$ P=4 a \int_{0}^{\pi / 2} \sqrt{1-e^{2} \cos ^{2} t} d t $$ where \(e\) is the eccentricity. (b) The integral in part (a) is called an elliptic integral. It has been studied at great length, and it is known that the integrand does not have an elementary antiderivative, so we must turn to approximate methods to evaluate \(P .\) Do so when \(a=1\) and \(e=\frac{1}{4}\) using the Parabolic Rule with \(n=4\). (Your answer should be near \(2 \pi\). Why?) AS (c) Repeat part (b) using \(n=20\).

Show that an ellipse and a hyperbola with the same foci intersect at right angles. Hint: Draw a picture and use the optical properties.

For what values of \(B\) is the graph of \(x^{2}+B x y+y^{2}=1\) (a) an ellipse (b) a circle (c) a hyperbola (d) two parallel lines

. Plot the Lissajous figure defined by \(x=\cos 2 t\), \(y=\sin 7 t, 0 \leq t \leq 2 \pi .\) Explain why this is a closed curve even though its graph does not look closed.

Sound travels at \(u\) feet per second and a rifle bullet at \(v>u\) feet per second. The sound of the firing of a rifle and the impact of the bullet hitting the target were heard simultaneously. If the rifle was at \(A(-c, 0)\), the target was at \(B(c, 0)\), and the listener was at \(P(x, y)\), find the equation of the curve on which \(P\) lies (in terms of \(u, v\), and \(c)\).

. Plot Lissajous figures for the following combinations of \(a\) and \(b\) for \(0 \leq t \leq 2 \pi\) : (a) \(a=1, b=2\) (b) \(a=4, b=8\) (c) \(a=5, b=10\) (d) \(a=2, b=3\) (e) \(a=6, b=9\) (f) \(a=12, b=18\)

Listeners \(A(-8,0), B(8,0)\), and \(C(8,10)\) recorded the exact times at which they heard an explosion. If \(B\) and \(C\) heard the explosion at the same time and \(A\) heard it 12 seconds later, where was the explosion? Assume that distances are in kilometers and that sound travels \(\frac{1}{3}\) kilometer per second.

Show that \(\left(\sqrt{x^{2}-a^{2}}-x\right) \rightarrow 0\) as \(x \rightarrow \infty\). Hint: Rationalize the numerator.

Plot the following parametric curves. Describe in words how the point moves around the curve in each case. (a) \(x=\cos \left(t^{2}-t\right), y=\sin \left(t^{2}-t\right)\) (b) \(x=\cos \left(2 t^{2}+3 t+1\right), y=\sin \left(2 t^{2}+3 t+1\right)\) (c) \(x=\cos (-2 \ln t), y=\sin (-2 \ln t)\) (d) \(x=\cos (\sin t), y=\sin (\sin t)\)

For an ellipse, let \(p\) and \(q\) be the distances from a focus to the two vertices. Show that \(b=\sqrt{p q}\), with \(2 b\) being the minor diameter.

. Using a computer algebra system, plot the following parametric curves for \(0 \leq t \leq 2 .\) Describe the shape of the curve in each case and the similarities and differences among all the curves. (a) \(x=t, y=t^{2}\) (b) \(x=t^{3}, y=t^{6}\) (c) \(x=-t^{4}, y=-t^{8}\) (d) \(x=t^{5}, y=t^{10}\)

Let \(P\) be a point on a ladder of length \(a+b, P\) being \(a\) units from the top end. As the ladder slides with its top end on the \(y\) -axis and its bottom end on the \(x\) -axis, \(P\) traces out a curve. Find the equation of this curve.

. Draw the graph of the epicycloid (see Problem 63 ) $$ \begin{array}{l} x=(a+b) \cos t-b \cos \frac{a+b}{b} t \\ y=(a+b) \sin t-b \sin \frac{a+b}{b} t \end{array} $$ for various values of \(a\) and \(b\). What conjectures can you make \((\) see Problem 73\() ?\)

Draw the Folium of Descartes \(x=3 t /\left(t^{3}+1\right)\), \(y=3 t^{2} /\left(t^{3}+1\right) .\) Then determine the values of \(t\) for which this graph is in each of the four quadrants.

If a horizontal hyperbola and a vertical hyperbola have the same asymptotes, show that their eccentricities \(e\) and \(E\) satisfy \(e^{-2}+E^{-2}=1\).