Chapter 10

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$r=\csc \theta$$

Describe the motion of a particle with position \((x, y)\) as \(t\) varies in the given interval. \(x=3+2 \cos t, \quad y=1+2 \sin t, \quad \pi / 2 \leqslant t \leqslant 3 \pi / 2\)

(a) Graph the conics \(r=e d /(1+e \sin \theta)\) for \(e=1\) and various values of \(d .\) How does the value of \(d\) affect the shape of the conic? (b) Graph these conics for \(d=1\) and various values of \(e\) . How does the value of \(e\) affect the shape of the conic?

Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. $$\frac{y^{2}}{16}-\frac{x^{2}}{36}=1$$

Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. $$x=\cos 3 \theta, \quad y=2 \sin \theta$$

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$r=\tan \theta \sec \theta$$

Describe the motion of a particle with position \((x, y)\) as \(t\) varies in the given interval. \(x=2 \sin t, \quad y=4+\cos t, \quad 0 \leqslant t \leqslant 3 \pi / 2\)

Show that a conic with focus at the origin, eccentricity \(e,\) and directrix \(x=-d\) has polar equation $$r=\frac{e d}{1-e \cos \theta}$$

Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. $$y^{2}-x^{2}=4$$

Use a graph to estimate the coordinates of the rightmost point on the curve \(x=t-t^{6}, y=e^{t}\) . Then use calculus to find the exact coordinates.

\(21-26\) Find a polar equation for the curve represented by the given Cartesian equation. $$x=3$$

Describe the motion of a particle with position \((x, y)\) as \(t\) varies in the given interval. \(x=5 \sin t, \quad y=2 \cos t, \quad-\pi \leqslant t \leqslant 5 \pi\)

Show that a conic with focus at the origin, eccentricity \(e,\) and directrix \(y=d\) has polar equation $$r=\frac{e d}{1+e \sin \theta}$$

Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. $$9 x^{2}-4 y^{2}=36$$

\(21-26\) Find a polar equation for the curve represented by the given Cartesian equation. $$x^{2}+y^{2}=9$$

Describe the motion of a particle with position \((x, y)\) as \(t\) varies in the given interval. \(x=\sin t, \quad y=\cos ^{2} t, \quad-2 \pi \leqslant t \leqslant 2 \pi\)

Show that a conic with focus at the origin, eccentricity \(e,\) and directrix \(y=-d\) has polar equation $$r=\frac{e d}{1-e \sin \theta}$$

Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. $$4 x^{2}-y^{2}-24 x-4 y+28=0$$

Graph the curve in a viewing rectangle that displays all the important aspects of the curve. $$x=t^{4}-2 t^{3}-2 t^{2}, \quad y=t^{3}-t$$

\(23-28\) Find the area of the region that lies inside the first curve and outside the second curve. $$r=2 \cos \theta, \quad r=1$$

\(21-26\) Find a polar equation for the curve represented by the given Cartesian equation. $$x=-y^{2}$$

Show that the parabolas \(r=c /(1+\cos \theta)\) and \(r=d /(1-\cos \theta)\) intersect at right angles.

Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. $$y^{2}-4 x^{2}-2 y+16 x=31$$

Graph the curve in a viewing rectangle that displays all the important aspects of the curve. $$x=t^{4}+4 t^{3}-8 t^{2}, \quad y=2 t^{2}-t$$

\(21-26\) Find a polar equation for the curve represented by the given Cartesian equation. $$x+y=9$$

The orbit of Mars around the sun is an ellipse with eccen- tricity 0.093 and semimajor axis \(2.28 \times 10^{8} \mathrm{km} .\) Find a polar equation for the orbit.

Identify the type of conic section whose equation is given and find the vertices and foci. $$x^{2}=y+1$$

Show that the curve \(x=\cos t, y=\sin t \cos t\) has two tangents at \((0,0)\) and find their equations. Sketch the curve.

\(23-28\) Find the area of the region that lies inside the first curve and outside the second curve. $$ r^{2}=8 \cos 2 \theta \quad r=2 $$

\(21-26\) Find a polar equation for the curve represented by the given Cartesian equation. $$x^{2}+y^{2}=2 c x$$

Jupiter's orhit has eccentricity 0.048 and the length of the major axis is \(1.56 \times 10^{9} \mathrm{km} .\) Find a polar equation for the orbit.

Identify the type of conic section whose equation is given and find the vertices and foci. $$x^{2}=y^{2}+1$$

\(21-26\) Find a polar equation for the curve represented by the given Cartesian equation. $$x y=4$$

The orbit of Halley's comet, last seen in 1986 and due to return in \(2062,\) is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is 36.18 \(\mathrm{AU}\) . [An astronomical unit (AU) is the mean distance between the earth and the sun, about 93 million miles.] Find a polar equa- tion for the orbit of Halley's comet. What is the maximum distance from the comet to the sun?

Identify the type of conic section whose equation is given and find the vertices and foci. $$x^{2}=4 y-2 y^{2}$$

(a) Find the slope of the tangent line to the trochoid \(x=r \theta-d \sin \theta, y=r-d \cos \theta\) in terms of \(\theta\) . (See Exercise 40 in Section \(10.1 . )\) (b) Show that if \(d\)<\(r\) then the trochoid does not have a vertical tangent.

\(23-28\) Find the area of the region that lies inside the first curve and outside the second curve. $$ r=3 \cos \theta, \quad r=1+\cos \theta $$

\(27-28\) For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve. (a) A line through the origin that makes an angle of \(\pi / 6\) with the positive \(x\) -axis (b) A vertical line through the point \((3,3)\)

The Hale-Bopp comet, discovered in \(1995,\) has an elliptical orbit with eccentricity 0.9951 and the length of the major axis is 356.5 AU. Find a polar equation for the orbit of this comet. How close to the sun does it come?

Identify the type of conic section whose equation is given and find the vertices and foci. $$y^{2}-8 y=6 x-16$$

(a) Find the slope of the tangent to the astroid \(x=a \cos ^{3} \theta\) \(y=a \sin ^{3} \theta\) in terms of \(\theta\) . (Astroids are explored in the Laboratory Project on page \(629 . )\) (b) At what points is the tangent horizontal or vertical? (c) At what points does the tangent have slope 1 or \(-1 ?\)

\(23-28\) Find the area of the region that lies inside the first curve and outside the second curve. $$ r=3 \sin \theta, \quad r=2-\sin \theta $$

\(27-28\) For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve. (a) A circle with radius 5 and center \((2,3)\) (b) A circle centered at the origin with radius 4

The planet Mercury travels in an elliptical orbit with eccen- tricity \(0.206 .\) Its minimum distance from the sun is \(4.6 \times 10^{7} \mathrm{km}\) . Find its maximum distance from the sun.

Identify the type of conic section whose equation is given and find the vertices and foci. $$y^{2}+2 y=4 x^{2}+3$$

At what points on the curve \(x=2 t^{3}, y=1+4 t-t^{2}\) does the tangent line have slope 1\(?\)

\(29-34\) Find the area of the region that lies inside both curves. $$ r=\sqrt{3} \cos \theta, \quad r=\sin \theta $$

Graph the curve \(x=y-3 y^{3}+y^{5}.\)

The distance from the planet Pluto to the sun is \(4.43 \times 10^{9} \mathrm{km}\) at perihelion and \(7.37 \times 10^{9} \mathrm{km}\) at aphelion. Find the eccentricity of Pluto's orbit.

Identify the type of conic section whose equation is given and find the vertices and foci. $$4 x^{2}+4 x+y^{2}=0$$