Identifying Conics by the Discriminant

by Dr. Carol JVF Burns (website creator)
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Conic sections—circles/ellipses, parabolas, hyperbolas—were introduced in the prior section.
They arise by intersecting a plane with the surface of an infinite double cone.
In this current section, we begin the analysis of equations representing conic sections.

There are results in this section for which you might be wondering: ‘Why is this true?’
If so, then you're encouraged to study the following optional sections:

General Conic Equation

Let $\,A\,,$ $\,B\,,$ $\,C\,,$ $\,D\,,$ $\,E\,$ and $\,F\,$ be real numbers. Consider the equation:

$$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \quad (\,\dagger\,) $$

The $\,x^2\,,$ $\,xy\,$ and $\,y^2\,$ terms are often called the degree-two terms in this equation.
Why? Because the exponents are $\,2\,,$ or they add to two:

$$\,xy = x^1y^1\ \ \ \ \text{ and } \ \ \ \ 1 + 1 = 2 $$

There is a beautiful relationship between equations of this type and the graphs of conic sections!
Indeed, both of the following are true:

For this reason, ($\,\dagger\,$) is called the general conic equation.
If the (nonzero) terms are in precisely the same order as indicated in ($\,\dagger\,$),
with zero on the right-hand side, then the general conic equation is said to be in standard form.

(By the way, the symbol ‘$\,\dagger\,$’ is a ‘dagger’, and is often used to mark an important equation.)

When you're ‘on the lookout’ for conics, you certainly don't need the equation to be in standard form!
Just check that there are no term types other than $\,x^2\,,$ $\,xy\,,$ $\,y^2\,,$ $\,x\,,$ $\,y\,$ and constants.
Any of the coefficients can equal zero, so you might not have all these term types.

It's not particularly easy to go from an arbitrary equation of the form ($\,\dagger\,$) to its graph, using paper-and-pencil.
(In the next few sections, you'll learn to graph fairly simple conics.)
However, WolframAlpha is up to the task!
If you're bored, jump up there, type in a random general conic equation, and see the awesomeness of WolframAlpha!

Here are some WolframAlpha results to illustrate the variety of conic sections (including some degenerate and limiting cases).
You can cut-and-paste the equations below each graphic into WolframAlpha yourself to verify the results. Enjoy!

5x^2-3xy+2y^2+3x-8y-7 = 0

(x-1)^2 + (y+3)^2 = 16

xy = 1

y = x^2 - 5x + 1

7 - 3xy - y = 9x^2 + y^2/4

x^2 + 2xy + y^2 = 0

x^2 + y^2 - 2xy = 5

x^2 - y^2 = 0

Or, you can play with the equation $\,Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\,$ below.
Use the sliders to set the desired values for the parameters $\,A\,,$ $\,B\,,$ $\,C\,,$ $\,D\,,$ $\,E\,,$ and $\,F\,.$
The initial configuration is $\,5x^2 - 3xy + 2y^2 + 3x - 8y - 7 = 0\,$; refresh the page to reset.
You can use the navigation arrows at the bottom right to see more of the graph.

Conic Discriminant

Even though conics can be challenging to graph, here's some really good news.
There's a simple way to decide which type of conic is represented by a given general conic equation, as follows:

The expression $\,B^2-4AC\,$ is called the discriminant because it helps us discriminate (differentiate between) the different types of conics.

Note the similarity to the discriminant of a quadratic equation $\,ax^2 + bx + c = 0\,.$
The quadratic discriminant is $\,b^2 - 4ac\,,$ and helps us decide how many real solutions the quadratic equation has.

EXAMPLES: Using the Discriminant to Determine the Type of Conic

Example 1: general conic equation in standard form

Consider the equation $\,5x^2 - 3xy + 2y^2 + 3x - 8y - 7 = 0\,$ (from the WolframAlpha example above).
It's already in the form $\,Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\,,$ so just read off the required coefficients

$$\begin{gather} A = 5\cr B = -3\cr C = 2 \end{gather} $$
and compute the discriminant
$$ B^2 - 4AC \ =\ (-3)^2 - 4(5)(2) \ =\ -31 \ < \ 0 $$
The discriminant is less than zero—it's an ellipse.

Example 2: general conic equation not in standard form

Consider the equation $\,y = x^2 - 5x + 1\,.$
It's a general conic equation, but it's not in standard form.

From prior mathematical experience, you know that quadratic functions graph as parabolas.
However, let's check that the ‘conic discriminant’ gives the same information.

You could put the equation in standard form: either

$$x^2 + 0xy + 0y^2 - 5x - y + 1 = 0$$
$$-x^2 + 0xy + 0y^2 + 5x + y - 1 = 0$$
In the first way, you'd read off $\,A = 1\,$ and $\,B = C = 0\,.$
In the second way, you'd read off $\,A = -1\,$ and $\,B = C = 0\,.$

But, there's an easier thought process.
Just check that the degree-two terms are all on the same side of the equation.
For the equation $\,y = x^2 - 5x + 1\,,$ there's only one degree-two term ($\,x^2\,$), so the side it's on doesn't matter.
So, $\,A = 1\,$ and $\,B = C = 0\,.$

In all cases, $\,B^2 - 4AC = 0\,.$
The discriminant equals zero—it's a parabola.

Example 3: a degenerate conic—a ‘point circle’

Consider the equation $\,x^2 + y^2 = 0\,.$
The only point that makes this equation true is $\,(x,y) = (0,0)\,.$

Read off $\,A = 1\,,$ $\,B = 0\,,$ and $\,C = 1\,.$
Thus, $\,B^2 - 4AC = 0^2 - 4(1)(1) = -4 < 0\,.$

Since the discriminant is negative, it's an ellipse.
Since, in addition, $\,A = C\,,$ it's a circle.
(A circle is a special case of an ellipse.)

This is a degenerate case: a circle of radius zero, fondly called a ‘point circle’.

Example 4: a hyperbola

Consider the equation $\displaystyle\,y = \frac 1x\,.$

In this equation, $\,x\,$ is not allowed to equal zero.
With $\,x \ne 0\,,$ multiplying both sides by $\,x\,$ gives $\,xy = 1\,,$ which is now recognizable as a conic.

In the equation $\,xy = 1\,,$ we have $\,A = C = 0\,$ and $\,B = 1\,.$
Thus, $\,B^2 - 4AC = 1 > 0\,.$

Since the discriminant is positive, it's a hyperbola.
The familiar reciprocal function is a hyperbola.

Important Notes

Here are additional things you should think about as you study this lesson:

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Intersecting an Infinite Double Cone and a Plane: Looking at the Equations

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
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(MAX is 27; there are 27 different problem types.)
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