Chapter 10

, find the length of the parametric curve defined over the given interval. $$ x=t, y=t^{3 / 2} ; 0 \leq t \leq 3 $$

Find the equation of the tangent line to the given curve at the given point. $$ \frac{x^{2}}{27}+\frac{y^{2}}{9}=1 \text { at }(3,-\sqrt{6}) $$

Ellipse with foci at \((2,0)\) and \((2,12)\) and a vertex at \((2,14)\)

Sketch the given curves and find their points of intersection. $$ r^{2}=4 \cos 2 \theta, r=2 \sqrt{2} \sin \theta $$

Prove that \(r=a \sin \theta+b \cos \theta\) represents a circle and find its center and radius.

, find the length of the parametric curve defined over the given interval. $$ x=2 \sin t, y=2 \cos t ; 0 \leq t \leq \pi $$

Find the equation of the tangent line to the given curve at the given point. $$ \frac{x^{2}}{2}-\frac{y^{2}}{4}=1 \text { at }(\sqrt{3}, \sqrt{2}) $$

For the parabola \(y^{2}=4 p x\) in Figure \(12, P\) is any of its points except the vertex, \(P B\) is the normal line at \(P, P A\) is perpendicular to the axis of the parabola, and \(A\) and \(B\) are on the axis. Find \(|A B|\) and note that it is a constant.

Parabola with focus \((2,5)\) and directrix \(x=10\)

Find the length of the latus rectum for the general conic \(r=e d /\left[1+e \cos \left(\theta-\theta_{0}\right)\right]\) in terms of \(e\) and \(d\).

, find the length of the parametric curve defined over the given interval. $$ x=3 t^{2}, y=t^{3} ; 0 \leq t \leq 2 $$

Find the equation of the tangent line to the given curve at the given point. $$ x^{2}+y^{2}=169 \text { at }(5,12) $$

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 \(a\) 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.

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 \(=\) \(2 e d / \sqrt{1-e^{2}}\)

, 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. $$ x^{2}-y^{2}=-1 \text { at }(\sqrt{2}, \sqrt{3}) $$

Show that the set of points equidistant from a circle and a line outside the circle is a parabola.

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.

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?

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

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.

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

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

In Problems \(43-48\), 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 path of a certain comet is a parabola with the sun at the focus. The angle between the axis of the parabola and a ray from the sun to the comet is \(120^{\circ}\) (measured from the point of the perihelion to the sun to the comet) when the comet is 100 million miles from the sun. How close does the comet get to the sun?

, 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. 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 \(0 \leq \theta \leq 10 \pi\)

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?

, 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. 4 x^{2}+x y+4 y^{2}=56

In order to graph a polar equation such as \(r=f(t)\) using a parametric equation grapher, you must replace this equation by \(x=f(t) \cos t\) and \(y=f(t) \sin t .\) These equations can be obtained by multiplying \(r=f(t)\) by \(\cos t\) and \(\sin t\), respectively. Confirm the discussions of conics in the text by graphing \(r=4 e /(1+e \cos t)\) for \(e=0.1,0.5,0.9,1,1.1\) and \(1.3\) on \([-\pi, \pi]\).

, 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 y-3 y^{2}=64

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 parametric curve defined over the given interval. $$ x=\sin t-t \cos t, y=\cos t+t \sin t ; \frac{\pi}{4} \leq t \leq \frac{\pi}{2} $$

Determine the length of the latus rectum (see Problem 45) of the hyperbola \(x^{2} / a^{2}-y^{2} / b^{2}=1\).

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. \(-\frac{1}{2} x^{2}+7 x y-\frac{1}{2} y^{2}-6 \sqrt{2} x-6 \sqrt{2} y=0\)

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. $$ r=\cos (13 \theta / 5) $$

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. \(\frac{3}{2} x^{2}+x y+\frac{3}{2} y^{2}+\sqrt{2} x+\sqrt{2} y=13\)

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. $$ r=\sin (5 \theta / 7) $$

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 $$ S=\int_{a}^{b} 2 \pi x \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t $$

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. $$ r=1+3 \cos (\theta / 3) $$

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.