Explanation 1) Concept: The polar equation of the form r = e d 1 ± e c o s θ o r r = e d 1 ± e s i n θ Represent a conic section with eccentricity e . The conic is an ellipse if e < 1 , a parabola if e = 1 or a hyperbola if e > 1 . 2) Given: r = 1 2 + s i n θ 3) Calculation: Consider the given polar equation r = 1 2 + s i n θ Multiply numerator and denominator by 1 2 r = 1 2 · 1 1 + 1 2 s i n θ Comparing the equation r = 1 · 1 2 1 + 1 2 s i n θ with r = e d 1 + e s i n θ get e = 1 2 e d = 1 2 ⇒ d = 1 The eccentricity is 1 2 . Conclusion: The eccentricity of the given curve is 1 2 . To identify: The given conic Solution: Ellipse. 1) Concept: The polar equation of the form r = e d 1 ± e c o s θ o r r = e d 1 ± e s i n θ Represent a conic section with eccentricity e . The conic is an ellipse if e < 1 , a parabola if e = 1 or a hyperbola if e > 1 . 2) Given: r = 1 2 + s i n θ 3) Calculation: From the part (a) e = 1 2 < 1 By using the concept, The given conic is ellipse. Conclusion: The given conic is ellipse. To give: The equation of the directrix. Solution: y = 1 1) Concept: If the polar equation of the form r = e d 1 + e s i n θ then equation of the directrix is y = d

8th Edition

ISBN: 9781285740621

Author: James Stewart

Publisher: Cengage Learning

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Chapter 10.6, Problem 10E

To determine

To find:

The eccentricity

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Chapter 10.6, Problem 9E

Chapter 10.6, Problem 11E

Chapter 10 Solutions

Calculus (MindTap Course List)

Ch. 10.1 - 14 Sketch the curve by using the parametric...Ch. 10.1 - 14 Sketch the curve by using the parametric...Ch. 10.1 - 14 Sketch the curve by using the parametric...Ch. 10.1 - 14 Sketch the curve by using the parametric...Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E

Ch. 10.1 - 1118 a Eliminate the parameter to find a Cartesian...Ch. 10.1 - 1118 a Eliminate the parameter to find a Cartesian...Ch. 10.1 - 1118 a Eliminate the parameter to find a Cartesian...Ch. 10.1 - 1118 a Eliminate the parameter to find a Cartesian...Ch. 10.1 - 1118 a Eliminate the parameter to find a Cartesian...Ch. 10.1 - 1118 a Eliminate the parameter to find a Cartesian...Ch. 10.1 - 1118 a Eliminate the parameter to find a Cartesian...Ch. 10.1 - 1118 a Eliminate the parameter to find a Cartesian...Ch. 10.1 - 1922 Describe the motion of a particle with...Ch. 10.1 - 1922 Describe the motion of a particle with...Ch. 10.1 - 1922 Describe the motion of a particle with...Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - Prob. 24E Ch. 10.1 - 2527 Use the graphs of x=f(t) and y=g(t) to sketch...Ch. 10.1 - 2527 Use the graphs of x=f(t) and y=g(t) to sketch...Ch. 10.1 - Prob. 27E Ch. 10.1 - Match the parametric equations with the graphs...Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - a Show that the parametric equations...Ch. 10.1 - Use a graphing device and the result of Exercise...Ch. 10.1 - Prob. 33E Ch. 10.1 - a Find parametric equations for the ellipse...Ch. 10.1 - Prob. 35E Ch. 10.1 - Prob. 36E Ch. 10.1 - Prob. 37E Ch. 10.1 - 3738 Compare the curves represented by the...Ch. 10.1 - Prob. 39E Ch. 10.1 - Prob. 40E Ch. 10.1 - Prob. 41E Ch. 10.1 - If a and b are fixed numbers, find parametric...Ch. 10.1 - A curve, called a witch of Maria Agnesi, consists...Ch. 10.1 - a Find parametric equations for the set of all...Ch. 10.1 - Suppose that the position of one particle at time...Ch. 10.1 - Prob. 46E Ch. 10.1 - Prob. 47E Ch. 10.1 - The swallowtail catastrophe curves are defined by...Ch. 10.1 - Prob. 49E Ch. 10.1 - Prob. 50E Ch. 10.1 - Prob. 51E Ch. 10.1 - Prob. 52E Ch. 10.2 - 12 Find dy/dx. x=t1+t,y=1+t Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - 36 Find and equation of the tangent to the curve...Ch. 10.2 - 36 Find and equation of the tangent to the curve...Ch. 10.2 - Prob. 7E Ch. 10.2 - Prob. 8E Ch. 10.2 - Prob. 9E Ch. 10.2 - Prob. 10E Ch. 10.2 - Prob. 11E Ch. 10.2 - Prob. 12E Ch. 10.2 - Prob. 13E Ch. 10.2 - Prob. 14E Ch. 10.2 - Prob. 15E Ch. 10.2 - Prob. 16E Ch. 10.2 - Prob. 17E Ch. 10.2 - Prob. 18E Ch. 10.2 - 1720 Find the points on the curve where the...Ch. 10.2 - Prob. 20E Ch. 10.2 - Prob. 21E Ch. 10.2 - Use a graph to estimate the coordinates of the...Ch. 10.2 - Prob. 23E Ch. 10.2 - Prob. 24E Ch. 10.2 - Prob. 25E Ch. 10.2 - Prob. 26E Ch. 10.2 - Prob. 27E Ch. 10.2 - a Find the slope of the tangent to the astroid...Ch. 10.2 - Prob. 29E Ch. 10.2 - Prob. 30E Ch. 10.2 - Use the parametric equations of an ellipse,...Ch. 10.2 - Prob. 32E Ch. 10.2 - Prob. 33E Ch. 10.2 - Prob. 34E Ch. 10.2 - Prob. 35E Ch. 10.2 - Let R be the region enclosed by the loop of the...Ch. 10.2 - Prob. 37E Ch. 10.2 - Prob. 38E Ch. 10.2 - Prob. 39E Ch. 10.2 - Prob. 40E Ch. 10.2 - Prob. 41E Ch. 10.2 - 4144 Find the exact length of the curve....Ch. 10.2 - Prob. 43E Ch. 10.2 - Prob. 44E Ch. 10.2 - 4546 Graph the curve and find its exact length....Ch. 10.2 - 4546 Graph the curve and find its exact length....Ch. 10.2 - Prob. 47E Ch. 10.2 - Prob. 48E Ch. 10.2 - Use Simpsons Rule with n=6 to estimate the length...Ch. 10.2 - Prob. 50E Ch. 10.2 - Prob. 51E Ch. 10.2 - Prob. 52E Ch. 10.2 - Show that the total length of the ellipse...Ch. 10.2 - Find the total length of the astroid...Ch. 10.2 - Prob. 55E Ch. 10.2 - Prob. 56E Ch. 10.2 - Prob. 57E Ch. 10.2 - 5760 Set up an integral that represents the area...Ch. 10.2 - Prob. 59E Ch. 10.2 - Prob. 60E Ch. 10.2 - Prob. 61E Ch. 10.2 - 6163 Find the exact area of the surface obtained...Ch. 10.2 - 6163 Find the exact area of the surface obtained...Ch. 10.2 - Prob. 64E Ch. 10.2 - Prob. 65E Ch. 10.2 - Prob. 66E Ch. 10.2 - If f is continuous and f(t)0 for atb, show that...Ch. 10.2 - Prob. 68E Ch. 10.2 - The curvature at a point P of a curve is defined...Ch. 10.2 - Prob. 70E Ch. 10.2 - Prob. 71E Ch. 10.2 - Prob. 72E Ch. 10.2 - A string is wound around a circle and then unwound...Ch. 10.2 - A cow is tied to a silo with radius r by a rope...Ch. 10.3 - Prob. 1E Ch. 10.3 - Prob. 2E Ch. 10.3 - Prob. 3E Ch. 10.3 - Prob. 4E Ch. 10.3 - Prob. 5E Ch. 10.3 - Prob. 6E Ch. 10.3 - Prob. 7E Ch. 10.3 - Prob. 8E Ch. 10.3 - Prob. 9E Ch. 10.3 - Prob. 10E Ch. 10.3 - Prob. 11E Ch. 10.3 - Prob. 12E Ch. 10.3 - Prob. 13E Ch. 10.3 - Prob. 14E Ch. 10.3 - Prob. 15E Ch. 10.3 - Prob. 16E Ch. 10.3 - Prob. 17E Ch. 10.3 - Prob. 18E Ch. 10.3 - 1520 Identify the curve by finding a Cartesian...Ch. 10.3 - 1520 Identify the curve by finding a Cartesian...Ch. 10.3 - Prob. 21E Ch. 10.3 - Prob. 22E Ch. 10.3 - Prob. 23E Ch. 10.3 - Prob. 24E Ch. 10.3 - Prob. 25E Ch. 10.3 - Prob. 26E Ch. 10.3 - Prob. 27E Ch. 10.3 - Prob. 28E Ch. 10.3 - Prob. 29E Ch. 10.3 - Prob. 30E Ch. 10.3 - Prob. 31E Ch. 10.3 - Prob. 32E Ch. 10.3 - Prob. 33E Ch. 10.3 - Prob. 34E Ch. 10.3 - Prob. 35E Ch. 10.3 - Prob. 36E Ch. 10.3 - Prob. 37E Ch. 10.3 - Prob. 38E Ch. 10.3 - Prob. 39E Ch. 10.3 - Prob. 40E Ch. 10.3 - Prob. 41E Ch. 10.3 - Prob. 42E Ch. 10.3 - Prob. 43E Ch. 10.3 - Prob. 44E Ch. 10.3 - Prob. 45E Ch. 10.3 - Prob. 46E Ch. 10.3 - Prob. 47E Ch. 10.3 - Prob. 48E Ch. 10.3 - Prob. 49E Ch. 10.3 - Prob. 50E Ch. 10.3 - Show that the curve r=sintan called a cissoid of...Ch. 10.3 - Prob. 52E Ch. 10.3 - a In Example 11 the graphs suggest that the limaon...Ch. 10.3 - Prob. 54E Ch. 10.3 - 5560 Find the slope of the tangent line to the...Ch. 10.3 - Prob. 56E Ch. 10.3 - Prob. 57E Ch. 10.3 - Prob. 58E Ch. 10.3 - Prob. 59E Ch. 10.3 - Prob. 60E Ch. 10.3 - Prob. 61E Ch. 10.3 - 6164 Find the points on the given curve where the...Ch. 10.3 - Prob. 63E Ch. 10.3 - Prob. 64E Ch. 10.3 - Prob. 65E Ch. 10.3 - Show that the curves r=asin and r=acos intersect...Ch. 10.3 - Prob. 67E Ch. 10.3 - Prob. 68E Ch. 10.3 - Prob. 69E Ch. 10.3 - Prob. 70E Ch. 10.3 - Prob. 71E Ch. 10.3 - Prob. 72E Ch. 10.3 - Prob. 73E Ch. 10.3 - Prob. 74E Ch. 10.3 - Prob. 75E Ch. 10.3 - Prob. 76E Ch. 10.3 - Prob. 77E Ch. 10.3 - Prob. 78E Ch. 10.4 - 14 Find the area of the region that is bounded by...Ch. 10.4 - 14 Find the area of the region that is bounded by...Ch. 10.4 - 14 Find the area of the region that is bounded by...Ch. 10.4 - Prob. 4E Ch. 10.4 - 58 Find the area of the shaded region. r2=sin2 Ch. 10.4 - 58 Find the area of the shaded region. r=2+cos Ch. 10.4 - 58 Find the area of the shaded region. r=4+3sin Ch. 10.4 - 58 Find the area of the shaded region. r=ln, 12 Ch. 10.4 - 912 Sketch the curve and find the area that it...Ch. 10.4 - 912 Sketch the curve and find the area that it...Ch. 10.4 - 912 Sketch the curve and find the area that it...Ch. 10.4 - 912 Sketch the curve and find the area that it...Ch. 10.4 - Prob. 13E Ch. 10.4 - Prob. 14E Ch. 10.4 - Prob. 15E Ch. 10.4 - Prob. 16E Ch. 10.4 - 1721 Find the area of the region enclosed by one...Ch. 10.4 - Prob. 18E Ch. 10.4 - 1721 Find the area of the region enclosed by one...Ch. 10.4 - 1721 Find the area of the region enclosed by one...Ch. 10.4 - 1721 Find the area of the region enclosed by one...Ch. 10.4 - Find the area enclosed by the loop of the...Ch. 10.4 - Prob. 23E Ch. 10.4 - 2328 Find the area of the region that lies inside...Ch. 10.4 - Prob. 25E Ch. 10.4 - Prob. 26E Ch. 10.4 - 2328 Find the area of the region that lies inside...Ch. 10.4 - Prob. 28E Ch. 10.4 - 2934 Find the area of the region that lies inside...Ch. 10.4 - 2934 Find the area of the region that lies inside...Ch. 10.4 - Prob. 31E Ch. 10.4 - Prob. 32E Ch. 10.4 - Prob. 33E Ch. 10.4 - Prob. 34E Ch. 10.4 - Prob. 35E Ch. 10.4 - Find the area between a larger loop and enclosed...Ch. 10.4 - Prob. 37E Ch. 10.4 - 3742 Find all points of intersection of the given...Ch. 10.4 - 3742 Find all points of intersection of the given...Ch. 10.4 - Prob. 40E Ch. 10.4 - Prob. 41E Ch. 10.4 - Prob. 42E Ch. 10.4 - Prob. 43E Ch. 10.4 - When recording live performances, sound engineers...Ch. 10.4 - Prob. 45E Ch. 10.4 - 4548 Find the exact length of the polar curve....Ch. 10.4 - 4548 Find the exact length of the polar curve....Ch. 10.4 - Prob. 48E Ch. 10.4 - 4950 Find the exact length of the curve. Use a...Ch. 10.4 - Prob. 50E Ch. 10.4 - Prob. 51E Ch. 10.4 - Prob. 52E Ch. 10.4 - Prob. 53E Ch. 10.4 - Prob. 54E Ch. 10.4 - a Use Formula 10.2 to show that the area of the...Ch. 10.4 - a Find a formula for the area of the surface...Ch. 10.5 - 18 Find the vertex, focus, and directrix of the...Ch. 10.5 - Prob. 2E Ch. 10.5 - Prob. 3E Ch. 10.5 - Prob. 4E Ch. 10.5 - 18 Find the vertex, focus, and directrix of the...Ch. 10.5 - Prob. 6E Ch. 10.5 - Prob. 7E Ch. 10.5 - Prob. 8E Ch. 10.5 - 910 Find an equation of the parabola. Then find...Ch. 10.5 - 910 Find an equation of the parabola. Then find...Ch. 10.5 - 1116 Find the vertices and foci of the ellipse and...Ch. 10.5 - Prob. 12E Ch. 10.5 - Prob. 13E Ch. 10.5 - 1116 Find the vertices and foci of the ellipse and...Ch. 10.5 - Prob. 15E Ch. 10.5 - Prob. 16E Ch. 10.5 - 1718 Find an equation of the ellipse. Then find...Ch. 10.5 - Prob. 18E Ch. 10.5 - Prob. 19E Ch. 10.5 - Prob. 20E Ch. 10.5 - Prob. 21E Ch. 10.5 - Prob. 22E Ch. 10.5 - Prob. 23E Ch. 10.5 - Prob. 24E Ch. 10.5 - Prob. 25E Ch. 10.5 - 2530 Identify the type of conic section whose...Ch. 10.5 - Prob. 27E Ch. 10.5 - Prob. 28E Ch. 10.5 - Prob. 29E Ch. 10.5 - Prob. 30E Ch. 10.5 - Prob. 31E Ch. 10.5 - Prob. 32E Ch. 10.5 - Prob. 33E Ch. 10.5 - 3148 Find an equation for the conic that satisfies...Ch. 10.5 - Prob. 35E Ch. 10.5 - Prob. 36E Ch. 10.5 - Prob. 37E Ch. 10.5 - Prob. 38E Ch. 10.5 - Prob. 39E Ch. 10.5 - Prob. 40E Ch. 10.5 - Prob. 41E Ch. 10.5 - Prob. 42E Ch. 10.5 - Prob. 43E Ch. 10.5 - Prob. 44E Ch. 10.5 - Prob. 45E Ch. 10.5 - Prob. 46E Ch. 10.5 - Prob. 47E Ch. 10.5 - Prob. 48E Ch. 10.5 - The point in a lunar orbit nearest the surface of...Ch. 10.5 - A cross-section of a parabolic reflector is shown...Ch. 10.5 - The LORAN LOng RAnge Navigation radio navigation...Ch. 10.5 - Use the definition of a hyperbola to derive...Ch. 10.5 - Prob. 53E Ch. 10.5 - Prob. 54E Ch. 10.5 - Prob. 55E Ch. 10.5 - Prob. 56E Ch. 10.5 - Prob. 57E Ch. 10.5 - Prob. 58E Ch. 10.5 - Prob. 59E Ch. 10.5 - Prob. 60E Ch. 10.5 - Prob. 61E Ch. 10.5 - Prob. 62E Ch. 10.5 - Prob. 63E Ch. 10.5 - a Calculate the surface area of the ellipsoid that...Ch. 10.5 - Let P(x1,y1) be a point on the ellipse...Ch. 10.5 - Let P(x1,y1) be a point on the hyperbola...Ch. 10.6 - Prob. 1E Ch. 10.6 - Prob. 2E Ch. 10.6 - Prob. 3E Ch. 10.6 - Prob. 4E Ch. 10.6 - Prob. 5E Ch. 10.6 - Prob. 6E Ch. 10.6 - Prob. 7E Ch. 10.6 - Prob. 8E Ch. 10.6 - Prob. 9E Ch. 10.6 - Prob. 10E Ch. 10.6 - Prob. 11E Ch. 10.6 - Prob. 12E Ch. 10.6 - Prob. 13E Ch. 10.6 - Prob. 14E Ch. 10.6 - Prob. 15E Ch. 10.6 - Prob. 16E Ch. 10.6 - Prob. 17E Ch. 10.6 - Prob. 18E Ch. 10.6 - Prob. 19E Ch. 10.6 - Prob. 20E Ch. 10.6 - Prob. 21E Ch. 10.6 - Prob. 22E Ch. 10.6 - Prob. 23E Ch. 10.6 - Prob. 24E Ch. 10.6 - Prob. 25E Ch. 10.6 - Prob. 26E Ch. 10.6 - The orbit of Halleys comet, last seen in 1986 and...Ch. 10.6 - Prob. 28E Ch. 10.6 - Prob. 29E Ch. 10.6 - Prob. 30E Ch. 10.6 - Prob. 31E Ch. 10.R - a What is a parametric curve? b How do you sketch...Ch. 10.R - Prob. 2CC Ch. 10.R - Prob. 3CC Ch. 10.R - Prob. 4CC Ch. 10.R - Prob. 5CC Ch. 10.R - Prob. 6CC Ch. 10.R - Prob. 7CC Ch. 10.R - a Give a definition of a hyperbola in terms of...Ch. 10.R - Prob. 9CC Ch. 10.R - Prob. 1TFQ Ch. 10.R - Prob. 2TFQ Ch. 10.R - Prob. 3TFQ Ch. 10.R - Prob. 4TFQ Ch. 10.R - Prob. 5TFQ Ch. 10.R - Prob. 6TFQ Ch. 10.R - Prob. 7TFQ Ch. 10.R - Prob. 8TFQ Ch. 10.R - Determine whether the statement is true or false....Ch. 10.R - Prob. 10TFQ Ch. 10.R - Prob. 1E Ch. 10.R - Prob. 2E Ch. 10.R - Prob. 3E Ch. 10.R - Prob. 4E Ch. 10.R - Prob. 5E Ch. 10.R - Prob. 6E Ch. 10.R - Prob. 7E Ch. 10.R - Prob. 8E Ch. 10.R - Prob. 9E Ch. 10.R - Prob. 10E Ch. 10.R - Prob. 11E Ch. 10.R - Prob. 12E Ch. 10.R - Prob. 13E Ch. 10.R - Prob. 14E Ch. 10.R - Prob. 15E Ch. 10.R - Prob. 16E Ch. 10.R - Prob. 17E Ch. 10.R - Prob. 18E Ch. 10.R - Prob. 19E Ch. 10.R - Prob. 20E Ch. 10.R - Prob. 21E Ch. 10.R - Prob. 22E Ch. 10.R - Prob. 23E Ch. 10.R - Prob. 24E Ch. 10.R - Prob. 25E Ch. 10.R - Prob. 26E Ch. 10.R - Prob. 27E Ch. 10.R - Prob. 28E Ch. 10.R - At what points does the curve...Ch. 10.R - Prob. 30E Ch. 10.R - Find the area enclosed by the curve r2=9cos5.Ch. 10.R - Prob. 32E Ch. 10.R - Prob. 33E Ch. 10.R - Prob. 34E Ch. 10.R - Prob. 35E Ch. 10.R - Find the area of the region that lies inside the...Ch. 10.R - 3740 Find the length of the curve. x=3t2,y=2t3,0t2 Ch. 10.R - Prob. 38E Ch. 10.R - 3740 Find the length of the curve. r=1/,2 Ch. 10.R - Prob. 40E Ch. 10.R - 4142 Find the area of the surface obtained by...Ch. 10.R - Prob. 42E Ch. 10.R - Prob. 43E Ch. 10.R - Prob. 44E Ch. 10.R - Prob. 45E Ch. 10.R - Prob. 46E Ch. 10.R - Prob. 47E Ch. 10.R - Prob. 48E Ch. 10.R - Prob. 49E Ch. 10.R - Find an equation of the parabola with focus (2,1)...Ch. 10.R - Prob. 51E Ch. 10.R - Prob. 52E Ch. 10.R - Prob. 53E Ch. 10.R - Prob. 54E Ch. 10.R - Prob. 55E Ch. 10.R - Prob. 56E Ch. 10.R - In the figure the circle of radius a is...Ch. 10.R - A curve called the folium of Descartes is defined...Ch. 10.P - The outer circle in the figure has radius 1 and...Ch. 10.P - a Find the highest and lowest points on the curve...Ch. 10.P - What is the smallest viewing rectangle that...Ch. 10.P - Four bugs are placed at the four corners of a...Ch. 10.P - Show that any tangent line to a hyperbola touches...Ch. 10.P - A circle C of radius 2r has its center at the...

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