Mathematics 101 Transitional Mathematics

Study Guide :: Unit 10

The Conics: A Special Case of Mathematical Relations

“Why do we have to study conics? I will never use it in my lifetime. Isn’t this a long outdated anachronism from the early Greeks? Maybe scientists used this at one time but we don’t use this anymore in our twenty-first century space age.”

Think again.

If you go to a Wikipedia web page on any of the planets in our solar system, each will be described in ultimate detail. For example, the one about our next door neighbour, Mars, gives the following data on its orbital characteristics:

If this sounds like Greek to you, it actually is. A semi-major axis is a geometric property of an ellipse and its length has a prominent place in the equation of the ellipse. This is how we can predict the positions of the planets at various times in the year. Aphelion and perihelion values give the location of the vertices of the elliptical orbit. Eccentricity values describe the ‘ovalness’ of the trajectory in its orbital plane. This ‘space age’ nomenclature still uses terms coined millennia ago.

Learning Objectives

After you have completed Unit 10, you should be able to:

1. give the geometric definitions of a parabola, an ellipse, a circle, and a hyperbola;
2. formulate both the standard and the general forms of the equation of an arbitrary parabola, ellipse, circle and hyperbola;
3. locate the positions and apply the formulas for the following: the vertex, the focus, the directrix, the axis of symmetry, the latus rectum, and the focal diameter of a parabola;
4. locate the positions and apply the formulas for the following: the centre, the two vertices, the two foci, the major axis, and the minor axis of an ellipse;
5. locate the positions and apply the formulas for the following: the centre, the two vertices, the two foci, the transverse axis, the conjugate axis, the central box, and the two slant asymptotes of a hyperbola;
6. describe the orientation and evaluate the eccentricity of any conic section;
7. draw a given conic section in standard position;
8. recognize and write the equations of conics which have been translated in the plane;
9. use the discriminant of a quadratic equation in two variables to identity its graph as a conic section; and
10. model and solve real-world situations using the conics sections.

Readings and Practice

Chapter 12 (pp. 421–477) of the textbook

Optional Sections

• Polar Equations of Conics (pp. 463–470)
• Focus on Modeling: Conics in Architecturepolar Equations of Conics (pp. 474–477)

Online graphing tool: Graph

Parabolas

Note that only the vertically oriented parabolas may be expressed as functions of a single variable ${\displaystyle x}$ (where the ${\displaystyle x}$-axis is the horizontal axis in the plane and the parabola’s axis of symmetry is the vertical ${\displaystyle y}$-axis). All of the other parabolas, including the horizontally oriented ones and the ones rotated out of their standard positions, are definable only as general relations in the two variables ${\displaystyle x}$ and ${\displaystyle y}$.

Textbook Readings

pp. 422–428

Practice

(pp. 428–430):  1, 2, 3, 4, 5, 7, 9, 11, 17, 19, 23, 27, 31, 35, 37, 39, 41, 45, 49, 53

Answers

pp. 545–546

Terms to Understand

geometric definition of a parabola; the vertex, the focus, the directrix, the axis of symmetry, the latus rectum, the focal diameter, the orientation, and the eccentricity of a parabola; the standard equation and the general equation of a parabola as a conic

Figures/Tables

Geometric Definition of a Parabola (p. 422)
Parabola with Vertical Axis (p. 423)
Parabola with Horizontal Axis (p. 424)

Online Tutorials—the Overview

Pertinent to “Parabolas”

• Conics: an introduction
• Parabolas in the real world
• Parabolas and their properties
• Equations of Parabolas III: Horizontally Aligned
• Equations of Parabolas IV: Vertically Aligned

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Ellipses

Note that all ellipses are expressible as mathematical relations in two variables, but not as functions of a single variable.

Textbook Readings

pp. 430–436

Practice

(pp. 436–439):  1, 2, 3, 4, 5, 7, 9, 11, 15, 21, 25, 27, 29 (try graphing by hand), 33, 35, 41, 45 (note that this is a nonlinear system of equations in two variables), 51

Answers

pp. 546–547

Terms to Understand

geometric definition of an ellipse; the centre, the two vertices, the two foci, the major axis, the minor axis, the orientation and eccentricity of an ellipse; the standard equation, and the general equation of an ellipse as a conic

Figures/Tables

Geometric Definition of an Ellipse (p. 430)
Ellipse with Centre at the Origin (p. 432)
Definition of Eccentricity (p. 435)

Online Tutorials—the Overview

Pertinent to “Ellipses”

• Conics: an introduction
• Ellipses in the real world
• Properties and Equations of Ellipses III: Horizontally Aligned
• Properties and Equations of Ellipses IV: Vertically Aligned
• Circles: a special kind of ellipse

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Hyperbolas

Note that all hyperbolas are expressible as mathematical relations in two variables, but not as functions of a single variable except when the slant asymptotes are the coordinate axes.

Textbook Readings

pp. 439–445

Practice

(pp. 445–447):  1, 2, 3, 4, 5, 7, 9, 15, 17, 21, 23, 25, 29 (try graphing by hand), 31, 37, 39, 41

Answers

p. 547

Terms to Understand

geometric definition of a hyperbola; the centre, the two vertices, the two foci, the transverse axis, the conjugate axis, the central box, the two slant asymptotes, the orientation and eccentricity of a hyperbola; the standard equation, and the general equation of a hyperbola as a conic

Figures/Tables

Geometric Definition of a Hyperbola (p. 439)
Hyperbola with Centre at the Origin (p. 440)
How to Sketch a Hyperbola (p. 441)

Online Tutorials—the Overview

Pertinent to “Hyperbolas”

• Conics: an introduction
• Hyperbolas in the real world
• Properties and Equations of Hyperbolas III: Horizontally Aligned
• Properties and Equations of Hyperbolas IV: Vertically Aligned

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Shifted Conics

The graphs of the conic sections can be shifted/translated in the plane in a similar way to the graphs of functions in the plane. As you proceed through this section, see if you can correlate the methods described for shifting equations to the methods of rigidly transforming functions learned in Unit 3. Are the procedures essentially the same or are they different?

Textbook Readings

pp. 448–453

Practice

(pp. 453–455):  1, 2, 3, 4, 7, 9, 13, 17, 19, 21, 27, 29, 31, 39

Answers

pp. 547–549

Terms to Understand

shifted conic

Figures/Tables

Shifting Graphs of Equations (p. 448)
General Equation of a Shifted Conic (p. 452)

Online Tutorials—the Overview

Pertinent to “Shifted Conics”

• Rigid transformations of functions: translations and reflections
• Non-rigid transformations of functions: expansions and contractions
• Composing transformations
• Equations of Parabolas III: Horizontally Aligned
• Equations of Parabolas IV: Vertically Aligned
• Properties and Equations of Ellipses III: Horizontally Aligned
• Properties and Equations of Ellipses IV: Vertically Aligned
• Properties and Equations of Hyperbolas III: Horizontally Aligned
• Properties and Equations of Hyperbolas IV: Vertically Aligned

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

The Discriminant

Recall that there is an expression, called the discriminant, associated with the general quadratic equation in one variable: ${\displaystyle a{x}^2+bx+c=0}$, namely ${\displaystyle b^2-4ac}$ provided ${\displaystyle a\ne 0}$. Also recall that the graphs of the corresponding quadratic functions ${\displaystyle p(x)=a{x}^2+bx+c}$ to these equations are all parabolas, provided ${\displaystyle a\ne 0}$. The sign of the discriminant turns out to be a rough indicator of where those parabolas are located in the plane, either above or below or touching or crossing the horizontal axis.

What is interesting is that there also is a discriminant associated with the general quadratic equation in two variables: ${\displaystyle A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0}$. The graphs of these equations turn out to be conic sections provided that at least one of ${\displaystyle A,B,C\ne 0}$. In this case, the sign of the discriminant, ${\displaystyle B^2-4AC}$, is an indicator as to which conic—parabola, ellipse, or hyperbola—the equation represents.

Textbook Readings

pp. 461–462

Practice

(pp. 462–463):  15a, 19a, 23a 27a, 31a

Answers

p. 549

Terms to Understand

discriminant of a quadratic equation in two variables

Figures/Tables

Identifying Conics through the Discriminant (p. 461)

Online Tutorials—the Overview

Pertinent to “The Discriminant”

• None

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Unit 10 Review

(pp. 470–472):  1–51 odd numbers, 55a, 57a

Answers

pp. 550–552

Unit 10 Test

(p. 473):  1–13a only

Answers

pp. 552–553

Figures/Tables

Formulas to Remember (pp. iv and v at the beginning of the text)

Final Exam Preparation

Units 1 through 10 with an emphasis on Units 6 through 10

Figures/Tables: Formulas to Remember (pp. iv and v at the beginning of the text)

Extra Practice

Take a random sample of questions from each of the unit sections below and try to solve them without looking at the text materials. That will give you some idea as to how well you have remembered and understood the concepts and procedures.

Pre-test for the Final Exam

Cumulative Review of Trigonometry

(pp. 398–399):  1–9, 12

Answers

p. 530

Cumulative Review Units 9 and 10

(pp. 478–479):  1–3, 7, 8a,

Answers

p. 553

Au Revoir

I do hope that you have learned a great deal about functions of a single real variable and how they can model our real world in so many diverse and even surprising ways. Many people, throughout almost 4500 years of collective human endeavour traversing both cultural and continental boundaries, have made our twenty-first century lives richer and, in some respects, even easier by their mathematical innovations and inventions. Here is a toast to your future endeavours.