Chapter 10

Show that the focal chord of the parabola \(y^{2}=4 p x\) with end points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) has length \(x_{1}+x_{2}+2 p\). Specialize to find the length \(L\) of the latus rectum.

Let \(r_{1}\) and \(r_{2}\) be the minimum and maximum distances (perihelion and aphelion, respectively) of the ellipse \(r=\) \(e d /\left[1+e \cos \left(\theta-\theta_{0}\right)\right]\) from a focus. Show that (a) \(r_{1}=e d /(1+e), r_{2}=e d /(1-e)\), (b) major diameter \(=2 e d /\left(1-e^{2}\right)\) and minor diameter \(=\) \(2 e d / \sqrt{1-e^{2}}\).

In Problems 35-46, find the length of the parametric curve defined over the given interval. $$ x=t+\frac{1}{t}, y=\ln t^{2} ; 1 \leq t \leq 4 $$

Find the equation of the tangent line to the given curve at the given point. \(x^{2}-y^{2}=-1\) at \((\sqrt{2}, \sqrt{3})\)

Find the equation of the given conic. Parabola with focus \((2,5)\) and vertex \((2,6)\)

Let \(a\) and \(b\) be fixed positive numbers and suppose that \(A P\) is part of the line that passes through \((0,0)\), with \(A\) on the line \(x=a\) and \(|A P|=b\). Find both the polar equation and the rectangular equation for the set of points \(P\) (called a conchoid) and sketch its graph.

The perihelion and aphelion for the orbit of the asteroid Icarus are 17 and 183 million miles, respectively. What is the eccentricity of its elliptical orbit?

In Problems 35-46, find the length of the parametric curve defined over the given interval. $$ x=2 e^{t}, y=3 e^{3 t / 2} ; \ln 3 \leq t \leq 2 \ln 3 $$

Let \(F\) and \(F^{\prime}\) be fixed points with polar coordinates \((a, 0)\) and \((-a, 0)\), respectively. Show that the set of points \(P\) satisfying \(|P F|\left|P F^{\prime}\right|=a^{2}\) is a lemniscate by finding its polar equation.

Earth's orbit around the sun is an ellipse of eccentricity \(0.0167\) and major diameter \(185.8\) million miles. Find its perihelion.

In Problems 35-46, find the length of the parametric curve defined over the given interval. $$ x=\sqrt{1-t^{2}}, y=1-t ; 0 \leq t \leq \frac{1}{4} $$

A line segment \(L\) of length \(2 a\) has its two end points on the \(x\) - and \(y\)-axes, respectively. The point \(P\) is on \(L\) and is such that \(O P\) is perpendicular to \(L\). Show that the set of points \(P\) satisfying this condition is a four-leaved rose by finding its polar equation.

Consider the parabola \(y=x^{2}\) over the interval \([a, b]\), and let \(c=(a+b) / 2\) be the midpoint of \([a, b], d\) be the midpoint of \([a, c]\), and \(e\) be the midpoint of \([c, b]\). Let \(T_{1}\) be the triangle with vertices on the parabola at \(a, c\), and \(b\), and let \(T_{2}\) be the union of the two triangles with vertices on the parabola at \(a, d, c\) and \(c\), \(e, b\), respectively (Figure 14). Continue to build triangles on triangles in this manner, thus obtaining sets \(T_{3}, T_{4}, \ldots\). (a) Show that \(A\left(T_{1}\right)=(b-a)^{3} / 8\). (b) Show that \(A\left(T_{2}\right)=A\left(T_{1}\right) / 4\). (c) Let \(S\) be the parabolic segment cut off by the chord \(P Q\). Show that the area of \(S\) satisfies $$ A(S)=A\left(T_{1}\right)+A\left(T_{2}\right)+A\left(T_{3}\right)+\cdots=\frac{4}{3} A\left(T_{1}\right) $$ This is a famous result of Archimedes, which he obtained without coordinates. (d) Use these results to show that the area under \(y=x^{2}\) between \(a\) and \(b\) is \(b^{3} / 3-a^{3} / 3\).

In Problems 35-46, find the length of the parametric curve defined over the given interval. $$ x=4 \sqrt{t}, y=t^{2}+\frac{1}{2 t}, \frac{1}{4} \leq t \leq 1 $$

A doorway in the shape of an elliptical arch (a halfellipse) is 10 feet wide and 4 feet high at the center. A box 2 feet high is to be pushed through the doorway. How wide can the box be?

Eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form. Finally, graph the equation showing the rotated axes. $$ x^{2}+x y+y^{2}=6 $$

Find the polar equation for the curve described by the following Cartesian equations. (a) \(y=45\) (b) \(x^{2}+y^{2}=36\) (c) \(x^{2}-y^{2}=1\) (d) \(4 x y=1\) (e) \(y=3 x+2\) (f) \(3 x^{2}+4 y=2\) (g) \(x^{2}+2 x+y^{2}-4 y-25=0\) Computers and graphing calculators offer a wonderful opportunity to experiment with the graphing of polar equations of the form \(r=f(\theta)\). In some cases these aids require that the equations be recast in a parametric form. Since \(x=r \cos \theta=f(\theta) \cos \theta\) and \(y=r \sin \theta=f(\theta) \sin \theta\), you can use the parametric graphing capabilities to graph \(x=f(t) \cos t\) and \(y=f(t) \sin t\) as a set of parametric equations.

The position of a comet with a highly eccentric elliptical orbit ( \(e\) very near 1 ) is measured with respect to a fixed polar axis (sun is at a focus but the polar axis is not an axis of the ellipse) at two times, giving the two points \((4, \pi / 2)\) and \((3, \pi / 4)\) of the orbit. Here distances are measured in astronomical units ( \(1 \mathrm{AU} \approx 93\) million miles). For the part of the orbit near the sun, assume that \(e=1\), so the orbit is given by $$ r=\frac{d}{1+\cos \left(\theta-\theta_{0}\right)} $$ (a) The two points give two conditions for \(d\) and \(\theta_{0}\). Use them to show that \(4.24 \cos \theta_{0}-3.76 \sin \theta_{0}-2=0\). (b) Solve for \(\theta_{0}\) using Newton's Method. (c) How close does the comet get to the sun?

In Problems 35-46, find the length of the parametric curve defined over the given interval. $$ x=\tanh t, y=\ln \left(\cosh ^{2} t\right) ;-3 \leq t \leq 3 $$

Eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form. Finally, graph the equation showing the rotated axes. $$ 3 x^{2}+10 x y+3 y^{2}+10=0 $$

Graph the curve \(r=\cos (8 \theta / 5)\) using the parametric graphing facility of a graphing calculator or computer. Notice that it is necessary to determine the proper domain for \(\theta\). Assuming that you start at \(\theta=0\), you have to determine the value of \(\theta\) that makes the curve start to repeat itself. Explain why the correct domain is \(0 \leq \theta \leq 10 \pi\).

In Problems 35-46, find the length of the parametric curve defined over the given interval. $$ x=\cos t, y=\ln (\sec t+\tan t)-\sin t ; 0 \leq t \leq \frac{\pi}{4} $$

How long is the latus rectum (chord through the focus perpendicular to the major axis) for the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) ?

Eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form. Finally, graph the equation showing the rotated axes. $$ 4 x^{2}+x y+4 y^{2}=56 $$

In Problems 35-46, find the length of the parametric curve defined over the given interval. $$ x=\sin t-t \cos t, y=\cos t+t \sin t ; \frac{\pi}{4} \leq t \leq \frac{\pi}{2} $$

In Problems 46-49, use a computer or graphing calculator to graph the given equation. Make sure that you choose a sufficiently large interval for the parameter so that the entire curve is drawn. \(r=\sqrt{1-0.5 \sin ^{2} \theta}\)

Find the length of the curve with the given parametric equations (a) \(x=\sin \theta, y=\cos \theta\) for \(0 \leq \theta \leq 2 \pi\) (b) \(x=\sin 3 \theta, y=\cos 3 \theta\) for \(0 \leq \theta \leq 2 \pi\) (c) Explain why the lengths in parts (a) and (b) are not equal.

Halley's comet has an elliptical orbit with major and minor diameters of \(36.18 \mathrm{AU}\) and \(9.12 \mathrm{AU}\), respectively ( \(1 \mathrm{AU}\) is 1 astronomical unit, the earth's mean distance from the sun). What is its minimum distance from the sun (assuming the sun is at a focus)?

Eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form. Finally, graph the equation showing the rotated axes. $$ -\frac{1}{2} x^{2}+7 x y-\frac{1}{2} y^{2}-6 \sqrt{2} x-6 \sqrt{2} y=0 $$

In Problems 46-49, use a computer or graphing calculator to graph the given equation. Make sure that you choose a sufficiently large interval for the parameter so that the entire curve is drawn. \(r=\cos (13 \theta / 5)\)

You can generate surfaces by revolving smooth curves, given parametrically, about a coordinate axis. As \(t\) increases from a to b, a smooth curve \(x=F(t)\) and \(y=G(t)\) is traced out exactly once. Revolving this curve about the \(x\)-axis for \(y \geq 0\) gives the surface of revolution with surface area $$ S=\int_{a}^{b} 2 \pi y \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t $$ See Section 6.4. Problems 48-54 relate to such surfaces. Derive a formula for the surface area generated by the rotation of the curve \(x=F(t), y=G(t)\) for \(a \leq t \leq b\) about the \(y\)-axis for \(x \geq 0\), and show that the result is given by $$ S=\int_{a}^{b} 2 \pi x \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t $$

Eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form. Finally, graph the equation showing the rotated axes. $$ \frac{3}{2} x^{2}+x y+\frac{3}{2} y^{2}+\sqrt{2} x+\sqrt{2} y=13 $$

In Problems 46-49, use a computer or graphing calculator to graph the given equation. Make sure that you choose a sufficiently large interval for the parameter so that the entire curve is drawn. \(r=\sin (5 \theta / 7)\)

A parametrization of a circle of radius 1 centered at \((1,0)\) in the \(x y\)-plane is given by \(x=1+\cos t, y=\sin t\), for \(0 \leq t \leq 2 \pi\). Find the surface area when this curve is revolved about the \(y\)-axis.

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 Problems 46-49, use a computer or graphing calculator to graph the given equation. Make sure that you choose a sufficiently large interval for the parameter so that the entire curve is drawn. \(r=1+3 \cos (\theta / 3)\)

Find the area of the surface generated by revolving the curve \(x=\cos t, y=3+\sin t\), for \(0 \leq t \leq 2 \pi\) about the \(x\)-axis.

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 \(49.31 \mathrm{AU}\). Find the major and minor diameters.

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 \(r=f(\theta-\alpha)\) ?

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 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 \(x=(2 / 3) t^{3 / 2}, y=2 \sqrt{t}\), 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.

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 \(x=t+\sqrt{7}, y=t^{2} / 2+\sqrt{7} t\), 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?

Polar graphs can be used to represent different spirals. The spirals can unwind clockwise or counterclockwise. Find the condition on \(c\) to make the spiral of Archimedes, \(r=c \theta\), unwind clockwise and counterclockwise.

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 $$ 3 x-3 \sqrt{2} y-7=0 $$