Instructional Unit on Conic Sections
This unit is designed as a set if investigations, along with background and historical information, to supplement Chapter 8 - Topics in Analytic Geometry of College Algebra and Trigonometry by Aufmann and Nation. (This is a two week unit in Math 115 - Precalculus Trigonometry at Macon College.) The figures were generated here by Geometer's Sketchpad and Algebra Xpresser. My plans are to amend the unit to use the TI-82, Maple and any other software that might be available to my class, the next time I teach this course.
INTRODUCTION
By definition, "A locus of points is a geometric figure containing all the points, and only those points, that satisfy a given condition or set of conditions." It may also be "the path if a moving point satisfying a given condition."
Descartes found that the graphs of second-degree equations in two variables always fall into one of seven categories:
[1] single point, [2] pair of straight lines, [3] circle, [4] parabola, [5] ellipse, [6] hyperbola, and [7] no graph at all.
We shall first look at the four loci: circle, ellipse, hyperbola, and parabola, known as non-degenerate conic sections from a geometric perspective.
I. Geometric Constructions:
Due to the nature of the course, this section will be a strictly informal investigation.
Let us start by constructing a circle. A circle is the set of points in a plane that are a fixed distance from a specified point. Instead of the usual compass or choice of circle off the GSP menu, start with a point "C" for the center (the focus), and a given segment CP. At random angles, draw segments CP, CP', CP", CP"', etc. around the point C.
Are you convinced yet that the P's all lie on a circle? Use the GSP feature to connect arc through 3 points around the circle to get all the P's connected.
An ellipse is the locus of points the sum of whose distance from two fixed points is constant. Identify the fixed points as A and B (these are the foci) and use GSP to create an ellipse. You may experiment on your own, or click here to bring up an ellipse that is already created. One example is shown below. You will find that as you move the B focus, the sum remains the same.
As you move the focus at B up, you will discover that the ellipse has become an hyperbola. Look at your sample measurement and see if that is consistent with the definition for hyperbola.
A parabola is the set of all points in a plane such that each point in the set is equidistant from a line called the directrix and a fixed point called the focus. Try to create a parabola using the definition, or click here to experiment with one that is already drawn. An example is shown below:
II. Algebraic - Rectangular
By definition: If A,B, C, D,E, and F are real numbers, and if at least one of A, B, and C is not 0, then
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
is called the general from of a second-degree equation in x and y.
Investigations:
1.) Let D = E = F = 0. Vary the values of A, B, and C to consider the following cases: B2 - 4AC < 0, B2 - 4AC = 0, and B2 - 4AC > 0. Identify which conics fit into each category.
For example: take B2 - 4AC < 0. Let A = [1,5], B = 0, c = [1,5]. As long as A and C have the same sign and B = 0, the conic is degenerate, it becomes a point.
2.) Reconsider each of the cases for B2 - 4AC, but this time vary the values of D, E, and F.
For example: take B2 - 4AC = 0. Let A = {-5, -4, -3, -2, -1, 1, 2, 3, 4, 5}, E = 2, B = C = D = F = 0. The result is a family of parabolas that are symmetrical about the y-axis.
Consider next the equation of the form
where (h,k) is the "center" and a and b are real number constants.
Investigations:
1.) Let (h,k) = (0,0) and a = b for various a's and b's. What type of figure do you get using a "+" and what type do you get with the "-"?
2.) Keep (h,k) = (0,0) but set a > b. What happens with "+" and "-" this time?
For example: X2 /a2 + y2/b2 = 1, for a = 2 and b = 1, then a = 5 and b = 2, and finally, a = 6 and b = 4.
We get a family of ellipses with the major axis on the x-axis.
3.) Now try the other cases and combinations of cases, i.e. a < b, move (h,k) off (0,0), and switch the x and y variable positions. Show figures and give explanations of the results.
III. Polar
The polar form of a conic is determined by a point, a directrix and a positive real number ratio known as the eccentricity.
Consider:
Investigations:
For each of the following investigations, let e = 1, e <1, e > 1.
1.) Vary p, first using the sum in the denominator, and then with the difference.
2.) What happens if you change the cosine to the sine while keeping the other variables fixed?
For example: For e = 1, p = 2, and "+" in the denominator.
Shown on the left below is: r = 2/(1 + sin x).
On the right below is: r = 2/(1 + cos x)
Both are parabolas. When the denominator contains the sine term, the directrix is horizontal. The cosine term determines a vertical directrix.
3.) You should have discovered that the above form only determines the parabola, hyperbola and ellipse. Write the equation for a circle using polar coordinates.
IV. Parametric Form
Recall the general definition for parametric equations:
Let t be a number in the interval I. A curve is the set of ordered pairs (x,y), where x = f(t) and y = g(t) for t, an element in I. The variable t is called a parameter, and the equations x = f(t) and y = g(t) are parametric equations.
Parametric equations are especially useful when describing a graph that is not a function. However, they can still be useful when the graph is a function.
Investigations:
1.) Consider x = a(sin t) and y = b(cos t) for various values of a and b, where a is not equal to b. Graph and describe the results.
2.) Consider the case above where a = b.
3.) Graph and discuss: x = t2 and y = t.
4.) This time amend # 3 so that the t's have coefficients, i.e. x = a (t2 ) and y = b(t). Try the cases with a = b, a < b, and a > b.
5.) Describe what happens when the variables swap to x = a(t) and y = b(t2).
6.) Now try to write a set of parametric equations to represent an hyperbola.
Further Fun:
1.) Graph y = (greatest integer of x)/(absolute value of x) over [-5,5].
Explain what you see, then look up equilateral hyperbolas and finish the explanation.
2.) Would a pool table in the shape of a circle or ellipse make sense? Why or why not?
Reading and Report Suggestions:
Many mathematical discoveries evolve over time. The Greek geometer Apollonius (c. 225 B.C.) wrote the definitive treatise on the investigation of the curves discussed here in Conic Sections. He is the one credited with introducing the terms "ellipse", "parabola", and "hyperbola". The original meaning compared the base of the geometric figure constructed to the length of a given segment. Ellipse meant defect, hyperbola meant excess, and parabola, "a placing beside".
The curves were of interest to the ancient geometers for use in trying to solve the construction problems of squaring a circle, doubling a square, and trisecting an angle. It took almost 1800 years for the mathematicians to see their practical use.
Choose a mathematician of topic from the following list for further study:
1.) Apollonius 7.) Menaechmus
2.) Archytas 8.) Pappus
3.) Descartes 9.) Pascal
4.) Euclid 10.) Pythagoras
5.) Hypatia 11.) Ramanujan
6.) Kepler
12.) doubling a square
13.) squaring a circle
14.) trisecting an angle
Ellipse applications:
15.) elliptical gears for machine tools
16.) optics
17.) orbit of a planet with the sun as a focus
18.) orbits of moons, satellites and comets
19.) ray emanating at one focus is reflected to the other
20.) whispering gallery
Hyperbola applications:
21.) comet path
22.) gear design
23.) navigation
24.) telescopic lenses
Parabola applications:
25.) antenna of a radio telescope
26.) cable of a suspension bridge
27.) flashlights
28.) parabolic reflector
29.) path of a projectile
30.) solar furnace