Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN: 9781305071742
Author: James Stewart, Lothar Redlin, Saleem Watson
Publisher: Cengage Learning
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Chapter 12.4, Problem 68E

DISCUSS: A Family of Confocal Conics Conics that share a focus are called confocal. Consider the family of conics that have a focus at ( 0 , 1 ) and a vertex at the origin, as shown in the figure.

(a) Find equations of two different ellipses that have these properties.

(b) Find equations of two different hyperbolas that have these properties.

(c) Explain why only one parabola satisfies these properties. Find it’s equation.

(d) Sketch the conics you found in parts (a),(b), and (c) on the same coordinate axes (for the hyperbolas, sketch top branches only).

(e) How are the ellipses and hyperbolas related to the parabola?

Chapter 12.4, Problem 68E, DISCUSS: A Family of Confocal Conics Conics that share a focus are called confocal. Consider the

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Chapter 12 Solutions

Algebra and Trigonometry (MindTap Course List)

Ch. 12.1 - A parabola is the set of all points in the plane...Ch. 12.1 - Prob. 2ECh. 12.1 - Prob. 3ECh. 12.1 - Prob. 4ECh. 12.1 - Prob. 5ECh. 12.1 - Prob. 6ECh. 12.1 - 5-10 Graphs of Parabolas Match the equation with...Ch. 12.1 - Prob. 8ECh. 12.1 - 5-10 Graphs of Parabolas Match the equation with...Ch. 12.1 - Prob. 10E
Ch. 12.1 - Prob. 11ECh. 12.1 - Prob. 12ECh. 12.1 - 11-24 Graphing Parabolas An equation of a parabola...Ch. 12.1 - 11-24 Graphing Parabolas An equation of a parabola...Ch. 12.1 - Prob. 15ECh. 12.1 - 11-24 Graphing Parabolas An equation of a parabola...Ch. 12.1 - Prob. 17ECh. 12.1 - Prob. 18ECh. 12.1 - 11-24 Graphing Parabolas An equation of a parabola...Ch. 12.1 - 11-24 Graphing Parabolas An equation of a parabola...Ch. 12.1 - Prob. 21ECh. 12.1 - Prob. 22ECh. 12.1 - Prob. 23ECh. 12.1 - 11-24 Graphing Parabolas An equation of a parabola...Ch. 12.1 - 25-30 Graphing Parabolas Use a graphing device to...Ch. 12.1 - Prob. 26ECh. 12.1 - Prob. 27ECh. 12.1 - Prob. 28ECh. 12.1 - Prob. 29ECh. 12.1 - Prob. 30ECh. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 32ECh. 12.1 - Prob. 33ECh. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 35ECh. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 38ECh. 12.1 - Prob. 39ECh. 12.1 - Prob. 40ECh. 12.1 - Prob. 41ECh. 12.1 - Prob. 42ECh. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 45ECh. 12.1 - 31-48 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 47ECh. 12.1 - Prob. 48ECh. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 51ECh. 12.1 - Prob. 52ECh. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - Prob. 54ECh. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - 49-58 Finding the Equation of a Parabola Find an...Ch. 12.1 - SKILLS 59-60 Families of Parabolasa Find equations...Ch. 12.1 - SKILLS 59-60 Families of Parabolas a Find...Ch. 12.1 - APPLICATIONS Parabolic Reflector A lamp with a...Ch. 12.1 - APPLICATIONS Satellite Dish A reflector for a...Ch. 12.1 - APPLICATIONS Suspension Bridge In a suspension...Ch. 12.1 - Prob. 64ECh. 12.1 - DISCUSS WRITE: Parabolas in the Real World Several...Ch. 12.1 - Prob. 66ECh. 12.2 - CONCEPTS An ellipse is the set of all points in...Ch. 12.2 - Prob. 2ECh. 12.2 - Prob. 3ECh. 12.2 - Prob. 4ECh. 12.2 - Prob. 5ECh. 12.2 - Prob. 6ECh. 12.2 - SKILLS 5-8Graphs of Ellipses Match the equation...Ch. 12.2 - Prob. 8ECh. 12.2 - Prob. 9ECh. 12.2 - Prob. 10ECh. 12.2 - Prob. 11ECh. 12.2 - Prob. 12ECh. 12.2 - SKILLS 9-28 Graphing Ellipse An equation of an...Ch. 12.2 - SKILLS 9-28 Graphing Ellipse An equation of an...Ch. 12.2 - Prob. 15ECh. 12.2 - SKILLS Graphing Ellipse An equation of an ellipse...Ch. 12.2 - Prob. 17ECh. 12.2 - Prob. 18ECh. 12.2 - SKILLS 9-28 Graphing Ellipses An equation of an...Ch. 12.2 - SKILLS 9-28 Graphing Ellipses An equation of an...Ch. 12.2 - Prob. 21ECh. 12.2 - Prob. 22ECh. 12.2 - Prob. 23ECh. 12.2 - Prob. 24ECh. 12.2 - 9-28. Graphing Ellipses An equation of ellipse is...Ch. 12.2 - 9-28. Graphing Ellipses An equation of ellipse is...Ch. 12.2 - Prob. 27ECh. 12.2 - Prob. 28ECh. 12.2 - Prob. 29ECh. 12.2 - Prob. 30ECh. 12.2 - 29-34 Finding the Equation of an Ellipse Find an...Ch. 12.2 - 29-34 Finding the Equation of an Ellipse Find an...Ch. 12.2 - Prob. 33ECh. 12.2 - Prob. 34ECh. 12.2 - Prob. 35ECh. 12.2 - Prob. 36ECh. 12.2 - 35-38 Graphing Ellipses Use a graphing device to...Ch. 12.2 - Prob. 38ECh. 12.2 - Prob. 39ECh. 12.2 - 39-56 Finding the Equation of an Ellipse Find an...Ch. 12.2 - Prob. 41ECh. 12.2 - Prob. 42ECh. 12.2 - 39-56 Finding the Equation of an Ellipse Find an...Ch. 12.2 - 39-56 Finding the Equation of an Ellipse Find an...Ch. 12.2 - Prob. 45ECh. 12.2 - Prob. 46ECh. 12.2 - Prob. 47ECh. 12.2 - 39-56 Finding the Equation of an Ellipse Find an...Ch. 12.2 - 39-56 Finding the Equation of an Ellipse Find an...Ch. 12.2 - Prob. 50ECh. 12.2 - Prob. 51ECh. 12.2 - Prob. 52ECh. 12.2 - Prob. 53ECh. 12.2 - SKILLS 39-56 Finding the Equation of an Ellipse...Ch. 12.2 - SKILLS 39-56 Finding the Equation of an Ellipse...Ch. 12.2 - Prob. 56ECh. 12.2 - Prob. 57ECh. 12.2 - Prob. 58ECh. 12.2 - Prob. 59ECh. 12.2 - Prob. 60ECh. 12.2 - SKILLS Plus Ancillary Circle The ancillary circle...Ch. 12.2 - Prob. 62ECh. 12.2 - Prob. 63ECh. 12.2 - How Wide Is an Ellipse at a Focus? A latus rectum...Ch. 12.2 - APPLICATIONS Perihelion and Aphelion The planets...Ch. 12.2 - Prob. 66ECh. 12.2 - APPLICATIONS Lunar Orbit For an object in an...Ch. 12.2 - Plywood Ellipse A carpenter wishes to construct an...Ch. 12.2 - Prob. 69ECh. 12.2 - Prob. 70ECh. 12.2 - Prob. 71ECh. 12.2 - DISCUSS Is It an Ellipse? A piece of paper is...Ch. 12.3 - CONCEPTS A Hyperbola is the set of all points in...Ch. 12.3 - Prob. 2ECh. 12.3 - Prob. 3ECh. 12.3 - Prob. 4ECh. 12.3 - Prob. 5ECh. 12.3 - Prob. 6ECh. 12.3 - SKILLS Graphs of Hyperbolas Match the equation...Ch. 12.3 - Prob. 8ECh. 12.3 - Prob. 9ECh. 12.3 - Prob. 10ECh. 12.3 - Prob. 11ECh. 12.3 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 12.3 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 12.3 - Prob. 14ECh. 12.3 - Prob. 15ECh. 12.3 - SKILLS 926Graphing HyperbolasAn equation of a...Ch. 12.3 - Prob. 17ECh. 12.3 - SKILLS 926Graphing Hyperbolas An equation of a...Ch. 12.3 - SKILLS 926Graphing Hyperbolas An equation of a...Ch. 12.3 - Prob. 20ECh. 12.3 - Prob. 21ECh. 12.3 - SKILLS 926Graphing Hyperbolas An equation of a...Ch. 12.3 - Prob. 23ECh. 12.3 - SKILLS 926Graphing Hyperbolas An equation of a...Ch. 12.3 - SKILLS 926Graphing Hyperbolas An equation of a...Ch. 12.3 - Prob. 26ECh. 12.3 - Prob. 27ECh. 12.3 - Prob. 28ECh. 12.3 - Prob. 29ECh. 12.3 - Prob. 30ECh. 12.3 - Prob. 31ECh. 12.3 - 27-32. Finding the equation of a Hyperbola Find...Ch. 12.3 - Prob. 33ECh. 12.3 - Prob. 34ECh. 12.3 - Prob. 35ECh. 12.3 - Prob. 36ECh. 12.3 - 37-50. Finding the Equation of a Hyperbola Find an...Ch. 12.3 - Prob. 38ECh. 12.3 - Prob. 39ECh. 12.3 - 37-50 Finding the equation of a HyperbolaFind the...Ch. 12.3 - Prob. 41ECh. 12.3 - Prob. 42ECh. 12.3 - 37-50 Finding the equation of a Hyperbola Find an...Ch. 12.3 - 37-50 Finding the equation of a Hyperbola Find an...Ch. 12.3 - Prob. 45ECh. 12.3 - Prob. 46ECh. 12.3 - Prob. 47ECh. 12.3 - Prob. 48ECh. 12.3 - 37-50 Finding the equation of a Hyperbola Find an...Ch. 12.3 - Prob. 50ECh. 12.3 - Prob. 51ECh. 12.3 - Prob. 52ECh. 12.3 - Prob. 53ECh. 12.3 - Prob. 54ECh. 12.3 - Prob. 55ECh. 12.3 - Prob. 56ECh. 12.3 - Comet Trajectories Some comets, such as Halleys...Ch. 12.3 - Prob. 58ECh. 12.3 - Prob. 59ECh. 12.3 - Prob. 60ECh. 12.4 - Suppose we want to graph an equation in x and y. a...Ch. 12.4 - Prob. 2ECh. 12.4 - Prob. 3ECh. 12.4 - Prob. 4ECh. 12.4 - Prob. 5ECh. 12.4 - Prob. 6ECh. 12.4 - SKILLS 5-12 Graphing Shifted Ellipses An equation...Ch. 12.4 - Prob. 8ECh. 12.4 - Prob. 9ECh. 12.4 - Prob. 10ECh. 12.4 - Prob. 11ECh. 12.4 - Prob. 12ECh. 12.4 - SKILLS 13-20 Graphing Shifted Parabolas An...Ch. 12.4 - Prob. 14ECh. 12.4 - Prob. 15ECh. 12.4 - Prob. 16ECh. 12.4 - Prob. 17ECh. 12.4 - Prob. 18ECh. 12.4 - SKILLS 13-20 Graphing Shifted Parabolas An...Ch. 12.4 - SKILLS 13-20 Graphing Shifted Parabolas An...Ch. 12.4 - Prob. 21ECh. 12.4 - Prob. 22ECh. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.4 - 21-28 Graphing Shifted HyperbolasAn equation of...Ch. 12.4 - Prob. 26ECh. 12.4 - Prob. 27ECh. 12.4 - Prob. 28ECh. 12.4 - Prob. 29ECh. 12.4 - Prob. 30ECh. 12.4 - Prob. 31ECh. 12.4 - Prob. 32ECh. 12.4 - Prob. 33ECh. 12.4 - Prob. 34ECh. 12.4 - Prob. 35ECh. 12.4 - Prob. 36ECh. 12.4 - 35-46Finding the Equation of a Shifted Conic Find...Ch. 12.4 - Prob. 38ECh. 12.4 - 35-46Finding the Equation of a Shifted Conic Find...Ch. 12.4 - Prob. 40ECh. 12.4 - Prob. 41ECh. 12.4 - Prob. 42ECh. 12.4 - Prob. 43ECh. 12.4 - Prob. 44ECh. 12.4 - Prob. 45ECh. 12.4 - Prob. 46ECh. 12.4 - Prob. 47ECh. 12.4 - Prob. 48ECh. 12.4 - SKILLS 47-58 Graphing Shifted Conics Complete the...Ch. 12.4 - Prob. 50ECh. 12.4 - Prob. 51ECh. 12.4 - Prob. 52ECh. 12.4 - Prob. 53ECh. 12.4 - Prob. 54ECh. 12.4 - 47-58 Graphing Shifted Conics Complete the square...Ch. 12.4 - SKILLS 47-58 Graphing Shifted Conics Complete the...Ch. 12.4 - Prob. 57ECh. 12.4 - Prob. 58ECh. 12.4 - Prob. 59ECh. 12.4 - Prob. 60ECh. 12.4 - 59-62. Graphing Shifting Conics Use a graphing...Ch. 12.4 - Prob. 62ECh. 12.4 - Prob. 63ECh. 12.4 - Prob. 64ECh. 12.4 - Prob. 65ECh. 12.4 - Prob. 66ECh. 12.4 - Orbit of a satellite A satellite is in an...Ch. 12.4 - DISCUSS: A Family of Confocal Conics Conics that...Ch. 12.5 - Suppose the x- and y-axes are routed through an...Ch. 12.5 - Prob. 2ECh. 12.5 - Prob. 3ECh. 12.5 - Prob. 4ECh. 12.5 - Prob. 5ECh. 12.5 - Prob. 6ECh. 12.5 - Prob. 7ECh. 12.5 - 3-8 Rotation of Axes Determine the XY-coordinates...Ch. 12.5 - Prob. 9ECh. 12.5 - Prob. 10ECh. 12.5 - Prob. 11ECh. 12.5 - Prob. 12ECh. 12.5 - Prob. 13ECh. 12.5 - Prob. 14ECh. 12.5 - Prob. 15ECh. 12.5 - Prob. 16ECh. 12.5 - Prob. 17ECh. 12.5 - Prob. 18ECh. 12.5 - Prob. 19ECh. 12.5 - Prob. 20ECh. 12.5 - Prob. 21ECh. 12.5 - 15-28 Graphing a Rotated Conic a Use the...Ch. 12.5 - Prob. 23ECh. 12.5 - Prob. 24ECh. 12.5 - Prob. 25ECh. 12.5 - Prob. 26ECh. 12.5 - Prob. 27ECh. 12.5 - Prob. 28ECh. 12.5 - Prob. 29ECh. 12.5 - Prob. 30ECh. 12.5 - Prob. 31ECh. 12.5 - Prob. 32ECh. 12.5 - Prob. 33ECh. 12.5 - Prob. 34ECh. 12.5 - Prob. 35ECh. 12.5 - Prob. 36ECh. 12.5 - Prob. 37ECh. 12.5 - Prob. 38ECh. 12.5 - Prob. 39ECh. 12.6 - CONCEPTS All conics can be described geometrically...Ch. 12.6 - Prob. 2ECh. 12.6 - Prob. 3ECh. 12.6 - Prob. 4ECh. 12.6 - Prob. 5ECh. 12.6 - Prob. 6ECh. 12.6 - Prob. 7ECh. 12.6 - Prob. 8ECh. 12.6 - Prob. 9ECh. 12.6 - Prob. 10ECh. 12.6 - Prob. 11ECh. 12.6 - Prob. 12ECh. 12.6 - Prob. 13ECh. 12.6 - Prob. 14ECh. 12.6 - Prob. 15ECh. 12.6 - Prob. 16ECh. 12.6 - Prob. 17ECh. 12.6 - Prob. 18ECh. 12.6 - Prob. 19ECh. 12.6 - Prob. 20ECh. 12.6 - Prob. 21ECh. 12.6 - Prob. 22ECh. 12.6 - Prob. 23ECh. 12.6 - Prob. 24ECh. 12.6 - Prob. 25ECh. 12.6 - Prob. 26ECh. 12.6 - Prob. 27ECh. 12.6 - Prob. 28ECh. 12.6 - Prob. 29ECh. 12.6 - Prob. 30ECh. 12.6 - Prob. 31ECh. 12.6 - Prob. 32ECh. 12.6 - Prob. 33ECh. 12.6 - Prob. 34ECh. 12.6 - Prob. 35ECh. 12.6 - Prob. 36ECh. 12.6 - Prob. 37ECh. 12.6 - Prob. 38ECh. 12.6 - Prob. 39ECh. 12.6 - Prob. 40ECh. 12.6 - Prob. 41ECh. 12.6 - Prob. 42ECh. 12.6 - Prob. 43ECh. 12.6 - Prob. 44ECh. 12.6 - Prob. 45ECh. 12.6 - Prob. 46ECh. 12.6 - Prob. 47ECh. 12.CR - Prob. 1CCCh. 12.CR - Prob. 2CCCh. 12.CR - a Give the geometric definition of a hyperbola. b...Ch. 12.CR - Prob. 4CCCh. 12.CR - Prob. 5CCCh. 12.CR - Prob. 6CCCh. 12.CR - Prob. 7CCCh. 12.CR - Prob. 8CCCh. 12.CR - Prob. 1ECh. 12.CR - Prob. 2ECh. 12.CR - Prob. 3ECh. 12.CR - Prob. 4ECh. 12.CR - Prob. 5ECh. 12.CR - Prob. 6ECh. 12.CR - Prob. 7ECh. 12.CR - Prob. 8ECh. 12.CR - Prob. 9ECh. 12.CR - Prob. 10ECh. 12.CR - Prob. 11ECh. 12.CR - Prob. 12ECh. 12.CR - Prob. 13ECh. 12.CR - Prob. 14ECh. 12.CR - Prob. 15ECh. 12.CR - Prob. 16ECh. 12.CR - Prob. 17ECh. 12.CR - Prob. 18ECh. 12.CR - Prob. 19ECh. 12.CR - Prob. 20ECh. 12.CR - Prob. 21ECh. 12.CR - Prob. 22ECh. 12.CR - Prob. 23ECh. 12.CR - Prob. 24ECh. 12.CR - Prob. 25ECh. 12.CR - Prob. 26ECh. 12.CR - Prob. 27ECh. 12.CR - Prob. 28ECh. 12.CR - Prob. 29ECh. 12.CR - Prob. 30ECh. 12.CR - Prob. 31ECh. 12.CR - Prob. 32ECh. 12.CR - Prob. 33ECh. 12.CR - Prob. 34ECh. 12.CR - Prob. 35ECh. 12.CR - Prob. 36ECh. 12.CR - Prob. 37ECh. 12.CR - Prob. 38ECh. 12.CR - Prob. 39ECh. 12.CR - Prob. 40ECh. 12.CR - Prob. 41ECh. 12.CR - Prob. 42ECh. 12.CR - Prob. 43ECh. 12.CR - Prob. 44ECh. 12.CR - Prob. 45ECh. 12.CR - Prob. 46ECh. 12.CR - Prob. 47ECh. 12.CR - Prob. 48ECh. 12.CR - Prob. 49ECh. 12.CR - Prob. 50ECh. 12.CR - 43-54 Indentifying and Graphic a conicDetermine...Ch. 12.CR - Prob. 52ECh. 12.CR - Prob. 53ECh. 12.CR - Prob. 54ECh. 12.CR - Prob. 55ECh. 12.CR - Prob. 56ECh. 12.CR - Prob. 57ECh. 12.CR - Prob. 58ECh. 12.CR - Prob. 59ECh. 12.CR - Prob. 60ECh. 12.CR - Prob. 61ECh. 12.CR - Prob. 62ECh. 12.CR - Prob. 63ECh. 12.CR - Prob. 64ECh. 12.CR - Prob. 65ECh. 12.CR - Prob. 66ECh. 12.CR - Prob. 67ECh. 12.CR - Prob. 68ECh. 12.CR - Prob. 69ECh. 12.CR - Prob. 70ECh. 12.CR - Prob. 71ECh. 12.CR - Prob. 72ECh. 12.CR - Prob. 73ECh. 12.CR - Prob. 74ECh. 12.CR - Prob. 75ECh. 12.CR - Prob. 76ECh. 12.CR - Prob. 77ECh. 12.CR - Prob. 78ECh. 12.CR - Prob. 79ECh. 12.CR - Prob. 80ECh. 12.CT - Prob. 1CTCh. 12.CT - Prob. 2CTCh. 12.CT - Prob. 3CTCh. 12.CT - Prob. 4CTCh. 12.CT - Prob. 5CTCh. 12.CT - Find an equation for the hyperbola with foci (0,5)...Ch. 12.CT - 7-9 Find an equation for the conic whose graph is...Ch. 12.CT - Prob. 8CTCh. 12.CT - Prob. 9CTCh. 12.CT - Prob. 10CTCh. 12.CT - 10-12 Determine whether the equation represents...Ch. 12.CT - Prob. 12CTCh. 12.CT - Prob. 13CTCh. 12.CT - Find an equation for the parabola with focus (2,4)...Ch. 12.CT - A parabolic reflector for a car headlight forms a...Ch. 12.CT - aUse the discriminant to determine whether the...Ch. 12.CT - Prob. 17CTCh. 12.FOM - Prob. 1PCh. 12.FOM - Prob. 2PCh. 12.FOM - Prob. 3PCh. 12.FOM - Prob. 4PCh. 12.FOM - Prob. 5PCh. 12.FOM - Prob. 6P
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