Conic Sections/Conic Sections Introduction

An Introduction

There exists a certain group of curves called Conic Sections that are conceptually kin in several astonishing ways. Each member of this group has a certain shape, and can be classified appropriately: as either a circle, an ellipse, a parabola, or a hyperbola. The term "Conic Section" can be applied to any one of these curves, and the study of one curve is not essential to the study of another. However, their correlation to each other is one of the more intriguing coincidences of mathematics.

But that's enough about conic sections. Perhaps you would instead like to view this short dialog between a mathematical novice and an expert about pi ($\pi$):

• Novice: So, what's pi?
• Expert: Pi a mathematical constant. It's always the same value, about 3.1415926. It's irrational and transcendental.
• Novice: Great. So what good is it?
• Expert: Well, the first thing that I can think of is for calculating volumes of surfaces of revolutions. That's calculus. And of course, radian measure.
• Novice: That's not what I meant. Why does it exist? Does it have a practical definition?
• Expert: Of course! It's the ratio of a circle's circumference to that circle's diameter. Pi a mathematical constant, you know. It's always the same value. I thought everybody knew that.
• Novice: (sardonically) Thanks.

This completely conic section-free piece of literature has some sort of application to conic sections. You may have caught that pi can be defined by a series of digits, but that doesn't help one use it, for there are plenty of numbers that can be defined by 3,1,4, and/or other numerals. Why is pi more useful than, for example, 2.7182818, that a letter of the Greek alphabet was chosen to represent it? On the other hand, any conic section can basically be defined with a general equation. On the other hand, most curves can be defined with a general equation. These equations do not get to the issue of what a conic section is any more than a string of digits actually defines pi.

Equations of Conic Sections

Nonetheless, here is just a sample of the sort of equations that are considered conic sections:

$y^{2}+8y=2x+4\,$
$y^{2}=3-x^{2}\,$
$x^{2}+4y^{2}-6x+16y=171\,$
${\frac {x^{2}}{9}}-{\frac {y^{2}}{49}}=1\,$

Browsing this list reveals several subtle similarities. For example, you might see that there are only several types of terms in these equations. A conic section is a quadratic curve, meaning that there are only two variables used, $x$ and $y$, and these variables only appear as themselves or squared, and possibly multiplied by a constant. One feature of conic sections is that they can be simplified down so that they only have these terms in them! (Another term, $x$ times $y$, can be used when rotating conic sections, but it's not relevant at the moment.)

Here are a couple of other equations for conic sections:
$(x-3)^{2}+(y+2)^{2}-16=0\,$
$(y-4)^{2}=2(x+4)\,$
$y=x^{2}+4\,$
$3x^{2}+3y^{2}-2xy=8\,$

The last equation above uses the xy term, which can be eliminated by a rotation. You'll notice that there are squared binomials (terms like $(x-3)^{2}$) in some of the equations. However, these equations can always be simplified so that $x^{2}$, $y^{2}$, $x$, $y$, and a constant are the only terms in the equation. This is a conic section's General Form, and looks like $Ax^{2}+Cy^{2}+Dx+Ey+F\,$, where A, C, D, E, and F are constants.

The use of squared binomials is one way to find graphical information about conic sections. A binomial is any algebraic expression that's the sum or difference of two terms. $y-4$, $5x^{2}-C$, and $ax+6$ are all binomials. The word binomial means "having two parts", hence the two different terms in a binomial.

Conic sections aren't all that different. Circles, ellipses, parabolas, and hyperbolas have several properties: the existence of foci, the locus of the intersection of a cone and a plane, and the consistency between the multiple definitions of eccentricity, including the ratio between the distance between foci and the distance between vertices, the slant of the plane, and the constant ratio between the distance from a focus to a directrix for any point on the curve. In addition, all conic sections are equations of the second degree or less. You basically can't have one characteristic and not have all the others.

We'll now conclude via the definition of conic sections from whence the term "conic section" derives it name.

A Conic Section Definition

Put simply, a conic section is a shape generated when a cone intersects with a plane. There are four main types of conic sections: parabola, hyperbola, circle, and ellipse. The circle is sometimes categorized as a type of ellipse. (See picture below)