Conic sections are the curves which can be derived from taking slices of a "double-napped" cone. A double-napped cone, in regular English, is two cones "nose to nose", with the one cone balanced perfectly on the other.
"Section" here is used in a sense similar to that in medicine or science, where a sample (from a biopsy, for instance) is frozen or suffused with a hardening resin, and then extremely thin slices (that is, sections) are shaved off for viewing under a microscope.
Content Continues Below
If you think of the double-napped cones as being hollow (say, made of cardboard), the curves we refer to as "conic sections" are the curves that result when we section, or slice, the cones at various angles. In other words, imagine taking that cardboard nose-to-nose cone construction and cutting across it with a hacksaw; those raw cut edges represent the curve which we call a conic section.
(To be a proper "section", you'd need to take a very thin slice of cardboard out of our construction, but that's not strictly necessary. The concept represented by that raw edge is what matters here.)
There are plenty of sites and books with pictures illustrating how to obtain the various curves through sectioning, so I won't bore you with more pictures here. And there are books and entire web sites devoted to the history of conics, the derivation and proofs of their formulas, and their various applications. I will not attempt to reproduce that information here.
This lesson, and the conic-specific lessons to which this page links, will instead concentrate on: finding curves, given points and other details; finding points and other details, given curves; and setting up and solving conics equations to solve typical word problems.
There are some basic terms that you should know for this topic:
You may encounter additional terms, depending on your textbook. Just make sure that you understand the particular terms that come up in your homework, so you're prepared for the test.
Content Continues Below
One very basic question that comes up pretty frequently is "Given an equation, how do I know which sort of conic it is?" Just as each conic has a typical shape:
...so also each conic has a "typical" equation form, often along the lines of the following:
parabola: Ax2 + Dx + Ey = 0
hyperbola: Ax2 − Cy2 + Dx + Ey + F = 0
circle: x2 + y2 + Dx + Ey + F = 0
ellipse: Ax2 + Cy2 + Dx + Ey + F = 0
These equations can be rearranged in various ways, and each conic has its own special form(s) that you'll need to learn to recognize, but some characteristics of the equations above remain unchanged for each type of conic.
If you keep these consistent characteristics in mind, then you can run through a quick check-list to determine what sort of conic is represented by a given quadratic equation.
Given a general-form conic equation in the form Ax2 + Cy2 + Dx + Ey + F = 0, or after rearranging to put the equation in this form (that is, after moving all the terms to one side of the "equals" sign), this is the sequence of tests that will tell you what type of curve will be graphed by a given conics equation:
No: It's a parabola.
Yes: Go to the next test....
Yes: It's an hyperbola.
No: Go to the next test....
Yes: It's a circle.
No: It's an ellipse.
A) 3x2 + 3y2 − 6x + 9y − 14 = 0
B) 6x2 + 12x − y + 15 = 0
C) x2 + 2y2 + 4x + 2y − 27 = 0
D) x2 − y2 + 3x − 2y − 43 = 0
A) Both variables are squared, and both squared terms are multiplied by the same number, so this is a circle.
B) Only one of the variables is squared, so this is a parabola.
C) Both variables are squared and have the same sign, but they aren't multiplied by the same number, so this is an ellipse.
D) Both variables are squared, and the squared terms have opposite signs, so this is an hyperbola.
If they give you an equation with variables on either side of the "equals" sign, rearrange the terms (on paper or in your head) to get the squared stuff together on one side. Then compare with the flow-chart above to find the type of equation you're looking at.
You can use the Mathway widget below to practice identifying conic equations (or skip the widget and continue below). Try the entered exercise, or type in your own exercise. Then click the button, scroll down, and select "Identify the Conic" to compare your answer to Mathway's.
(Click "Tap to view steps" to be taken directly to the Mathway site for a paid upgrade.)
You may have noticed, in the table of "typical" shapes (above), that the graphs paralleled either the x-axis or the y-axis, and you may have wondered whether conics can ever be "slanted" with respect to the axes.
As you can see below, yes, conics can have graphs that don't parallel the x- or y-axes, but instead go off at angles to the axes:
But the equations for the "slanty" conics get so much more messy that you can't deal with them until after trigonometry. If you wondered why the coefficients in the "general conic" equations, such as Ax2 + Cy2 + Dx + Ey + F = 0, skipped the letter B, it's because the B is the coefficient of the "xy" term that you can't handle until after you've taken trig.
You'll probably never have to deal with the "slanty" conics until calculus, when you may have to do "rotation of axes". Don't be in a rush. It's not a pretty topic.
Technically, yes. Think about the two nose-to-nose cones one uses for conics, and slice those cones so that the cut passes through the point where the two cones' tips touch. The curve created is just straight lines that pass along the outsides of the cones.
Once you have classified a conic, what can you do with it? The following lessons give some examples: