Conics as a Slice of a Cone
The Greeks originally conceived of these as the shapes generated by slicing through a right circular cone. In A History of Greek Mathematics Vol. II (1921), Sir Thomas Heath says:
The question arises, how did Menaechmus come to think of obtaining curves by cutting a cone? On this we have no information whatever. (pg. 110)Lost in the mists of time!
More after the jump...
Conics on the Cartesian Plane
In terms of Cartesian analytic geometry, each conic has its respective equation and completing the square is an essential element to this conception of the conics. Each of these general equations requires algebraic perfect squares to analyze the position and shape of the curve. In the case of the parabola, (h,k) represents the vertex, for the circle, ellipse and hyperbola, (h,k) is the center.
Circle: (x-h)^2 + (y-k)^2 = r^2
Ellipse: (x-h)^2/a^2 + (y-k)^2/b^2 = 1
Hyperbola: (x-h)^2/a^2 - (y-k)^2/b^2 = 1
As I sometimes mention to my students, these "standard" equations sometimes come in different forms in different textbooks. Some texts isolate the y variable in the Parabola equation to arrive at
In many situations it's useful to orient the parabola with a vertex at the origin as:
I typically use this last form for application problems, but use the more general form in studying the general parabola. It's actually interesting that you can square out the general equation of the parabola and collect like terms to show that -b/2a in the quadratic y=ax^2+bx+c does return a value of "h" as the x coordinate of the vertex.
Conics as a Locus of Points
The conics are also defined as loci - that is, as a locus (or collection) of points that have a particular property:
Parabola: a locus of points equidistant from a given point (focus) and a given line (directrix).
Circle: a locus of points equidistant from a given point (center of the circle).
Ellipse: a locus of points the sum of whose distances from two given points (foci) is constant.
Hyperbola: a locus of points the difference of whose distances from two given points (foci) is constant.
The analytic equations can be derived from the locus definitions, mainly by using the distance formula.
Something I have always found amazing about the conics is their appearance in nature. The parabola is an easy one to show as any object thrown into the air traces out a parabola (gravity's rainbow). This can be proven using a little physics and trigonometry:
In the picture above the straight line from lower left to upper right is the path an object would take without gravity (in a vacuum), and the dangling pieces show the position of the object as gravity acts on it, creating the parabolic path. The length of each string in the picture is the height lost due to gravity.
I've done application problems on parabolic trajectory, but in the Pre-Calculus course I currently teach, we focus on the paraboloid of revolution that is used in satellite dishes, headlights and flashlights. These applications depend on the amazing property that all light and radio waves hitting a parabolic dish are reflected to the focus of the dish. Headlights and flashlights use the opposite property that all light emanating from the focus will be reflected off the paraboloid and shine forward in a concentrated beam. I also sometimes mention the parabolic microphone often used in televising football games
The paraboloid shape is also being used to generate electricity from solar radiation. A tower containing salt is placed at the focus and the sun's heat energy is used to melt the salt and heat it to 500 degrees Celsius. The molten salt then runs through a pipe in the middle of a water tank generating steam to drive a turbine.
The other conic application that I find fascinating is the work of Kepler, Brahe and Copernicus that showed that the planets' paths about the sun are elliptical. The Greeks conceived of the ellipse before 300 BCE and Kepler did his work right around 1600 CE - a span of nearly 2000 years between conception and application.
The hyperbola can be used to describe the path a comet might take when passing through our solar system and is also used in the LORAN navigation system to describe the intersection of radio waves.
I generally have about 7 class periods in which to cover the conics, so I focus on completing the square, the general equations for the circle and parabola and applications with the paraboloid.
Here's an interesting project related to the parabola that was inspired by Paul Nahin's book An Imaginary Tale.
Sullivan covers the circle in Section 2.4, with the rest of the conics in Chapter 11. He gives the equations in a similar form as I've listed above. Coburn gives the same versions of these equations for the circle, ellipse and hyperbola, but goes through some fairly convoluted explanations of what is happening with the equation for the parabola. He ends up presenting the y=ax^2+bx+c, the x^2=4py, and the (x-h)^2=4p(y-k) all together in the same section with appropriate explanations, but I think that this might end up being something of a jumble for a student (or unprepared teacher).
Sobel and Lerner cover the circle in section 4.1 and then cover the ellipse, hyperbola and parabola in chapter 10 (together with non-linear systems). They give the equations in a similar form as I've listed above.
Paul Foerster (typically) is a little different from everyone else - he covers the conics in Chapter 12 with the most general equation of all -
Ax^2+Bxy+Cy^2+Dx+Ey+F=0I've left out this form of the conic equations because I've never had the opportunity to teach it before. The xy term rotates a particular conic away from the vertical (or horizontal) orientation it would otherwise have. He follows this up with equations similar to the ones in Sullivan and Coburn but given in an odd form with a "scaling factor" which I've never seen used before.
Brown and Robbins cover the circle in Section 1-6 and the parabola in the form y=ax^2+bx+c in Section 2-3, with the general conics in Chapter 11. His gives his ellipse and hyperbola equations in a similar form as the other texts (except Foerster) but uses the (y-k)=1/(4p)*(x-h)^2 for the parabola.
Dolciani covers the conics in Chapter 13, first presenting the curves with center (or vertex) at the origin, and then using the more general equations to introduce some of the major ideas of transformations in the plane. She ends with the most general form Ax^2+Bxy+Cy^2+Dx+Ey+F=0.