Algebra 2 Conic Sections

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In the past four lectures, we have discussed various conic sections: parabolas, circles, ellipses, and hyperbolas.0003

And this lecture is designed to bring that information together and to give you some context about this.0010

First of all, what are conic sections? We know we can name them; we know what they are; but where do they come from?0016

Well, they are literally sections of a cone: when you take a double cone (it is actually a double cone, as follows, with the points together),0023

and you section them (sectioning is slicing)--when you take slices of them, using a plane, you come up with these four types of curves.0034

So, as you can see, when you take a plane and section, or slice, the cone across, you are going to end up with a circle.0043

If you tip that plane at an angle, the result is an ellipse.0054

If you encompass the edge of the come, you end up with a parabola.0065

And if you slice through in such a way that you capture the edges of both cones, then you end up with a hyperbola; and there you can see the two branches.0071

So, this is where conic sections come from; and they have many applications in science.0079

We have talked about the standard form of each conic section (for example, the standard form of a circle, or the standard form of an ellipse).0087

This standard form that I am talking about now is a very general form.0095

It gives you a general equation, ax² + bxy + cy² + dx + ey + f = 0.0098

So, what we are going to do in a minute is talk about how you can look at this general form and determine0109

which type of conic section you are working with, so that you can put the equation0115

in the standard form particular to that type of conic section.0119

And as we have been going through, I have mentioned some ways that you can tell, if you just have an equation in the general form,0124

what type of conic section you are working with; and now I am going to bring that all together.0131

OK, if b = 0, we can analyze that standard form of the conic section to determine what type of conic section the equation represents.0137

Looking back at that general standard form again, ax² + bxy + cy² + dx + ey + f = 0.0146

Here, we are having the limitation that b = 0.0168

And throughout this course, when we work with conic sections, we have only worked with ones where b is 0.0171

When b is 0, you end up with this.0178

Once you have this standard form, then you can go ahead and analyze it in ways we are going to discuss in a minute0192

to determine which type of conic section you have (what the equation describes).0197

But let's talk for a minute about what the xy tells you.0201

So far, we have worked with shapes such as parabolas; and some were oriented vertically; some, horizontally.0204

We also worked with ellipses (some had a horizontal major axis; some had a vertical major axis); and the same with hyperbolas.0215

So, even though the center may have been shifted, these were all either strictly vertical or strictly horizontal.0230

What this bxy term does is rotates it so that instead of, say, having an ellipse that has0237

a completely vertical major axis or horizontal major axis, you could end up with an ellipse like this--0243

the major axis is at an angle--or a parabola that is like this.0250

And that is definitely more complicated to work with; and it doesn't allow us to complete the square, then,0259

to shift an equation from the general form to a specific standard form.0265

So, later on, if you continue on in math, you may end up working with these shapes.0270

But for this course, we are limiting it to conic sections that are either vertical or horizontal; but they are not tipped at any other type of angle.0274

In order to identify conic sections, you need to look at the coefficients of the x² and y² terms.0286

So, let's rewrite this; and again, the assumption is that b = 0.0293

So, I am just going to have ax² + cy² + dx + ey + f = 0.0299

Parabola: Recall that, with a parabola, you have an x² term or a y² term, but not both.0310

Therefore, either a is 0 (so this drops out) or c is 0 (so this drops out).0325

An example would be something like x = 3y² + 2y + 6.0331

Or you might have y = 2x² - 4x + 8.0338

So, neither of these has both an x² and a y² term in the same equation.0344

For a circle, recall that what you are going to end up with is an x² and a y² term0352

on the same side of the equation, with the same sign; and they are going to have the same coefficients.0361

Therefore, a is going to equal c.0366

An example would be x² + y² + 3x - 5y - 10 = 0.0369

Here, a equals 1, and c equals 1; those are the same coefficients; x² and y²0378

have the same sign and the same side of the equation; so it is a circle.0386

If we are working with an ellipse, this time the x² term and y² terms are going to be0392

on the same side of the equation, with the same sign (like with the circle), but a and c are different.0399

They are unequal; that tells me that I am working with an ellipse.0405

For example, 12x² + 9y² + 25x + 28y + 40 = 0.0409

Here, I have a = 12 and c = 9; so this is the equation describing an ellipse.0423

Finally, with a hyperbola, these are pretty straightforward to recognize, because you are going to have0431

an x² term and a y² term, but they are going to have opposite signs.0436

Their coefficients will have opposite signs.0440

For example, 4x² - 8y² + 10x + 6y - 34 = 0.0443

So, I have a = 4 and c = -8; since a and c have opposite signs, this is an equation describing a hyperbola.0458

You can use these rules to allow you to identify conic sections when you are given an equation in what we are going to call "general form."0467

It is standard form, but it is a very general standard form for any type of conic section.0476

OK, now we are going to work on identifying the various conic sections by looking at their equations.0481

First, write in standard form, and identify the conic section.0488

OK, so general standard form is what I am talking about right now: it is x² + 2y².0495

I need to subtract 4x from both sides, subtract 12, and set everything equal to 0.0503

What this tells me is that I have a = 1 and c = 2.0510

Since a = 1 and c = 2, these have the same sign (the x² and the y² terms); but they have different coefficients.0515

And that means that what I am working with is an ellipse.0525

You could go on, then, and write this in the specific standard form for an ellipse.0531

Let's do that by completing the square: start out by grouping...let's rewrite it here.0537

And then, let's group the x and the y terms; so x² terms group together; y terms group together.0545

And now, add 12 to both sides to move that over, to make completing the square a little bit easier.0555

To complete the square for x² - 4x, I need to add b²/4.0563

b²/4 is equal to 4²/4, is 16/4; it is 4.0571

So, I add x² - 4x + 4; and it is very important to remember to add the 4 to the right side, as well.0582

There is no factor out here; I don't need to multiply--it is just 1; so 4 times 1 is 4; that gives me 12 + 4.0592

All right, that is x² - 4x + 4 + 2y² = 16.0601

This can be rewritten as (x - 2)² + 2y² = 16.0609

But recall, in standard form for an ellipse, you need to have a 1 on the right.0616

So, rewrite this up here, and then divide both sides by 16.0622

This is just (x - 2)²/16; this will cancel; this will become 8; and then 16/16 is 1.0633

So, we started out with this equation, put it into the general standard form to identify that this is an ellipse,0645

and then went on to complete the square; and now I have it written in standard form specifically for an ellipse,0652

which is much more useful when you are working with that and trying to graph.0657

This time, without completing the square, all we are going to do is identify the conic section.0664

And this is already in standard form; therefore, a = 2; c = -3.0668

Since a and c have opposite signs, this is the equation for a hyperbola.0678

I have an x² term and a y² term, both, so it is not a parabola.0687

They have opposite signs; therefore, it must describe a hyperbola.0691

OK, write in standard form and identify the conic section.0698

Right now, this is not in any type of standard form; so I am going to work with the general standard form.0702

First, I am going to subtract 36x² from both sides.0708

Then, I am going to subtract 128 from both sides.0717

This means that I have a = -36, and c = 16; since these two are opposite signs, this is an equation describing a hyperbola.0727

OK, now, let's go ahead and put this in standard form specific to a hyperbola.0741

And let's start out by moving this 128 back over to the right; this is actually 32.0747

Next, I do have a common factor of 4, so I am going to divide both sides by that, so that I am working with smaller numbers.0763

That is -9x² + 4y² + 8y =...128/4 would give 32.0770

All right, now to make this already move it more towards looking like a hyperbola, I am going to put the positive terms here in front:0786

4y² + 8y - 9x², because I am going to have a difference.0793

To complete the square, I first need to factor out that 4; then I need to add b²/4 to this expression.0801

This is going to equal 2²/4; that is 4/4, which is 1.0816

Here is where I need to be careful, because I need to make sure I add 4 times 1 to the right, which is 4, to keep the equation balanced.0825

At this point, I am going to rewrite this as (y + 1)² - 9x² = 36.0837

The last step is: I want the right side to be 1, so I am going to divide both sides by 36.0844

4 goes into 36 nine times; 9 goes into 36 four times; and then this cancels out to 1.0860

OK, so I started out with an equation that wasn't in any kind of standard form.0872

I put it in general standard form, and then determined it was a hyperbola, completed the square, and ended up0876

with an equation in the standard form for a hyperbola, so that I can use that to graph the hyperbola, if needed.0882

Write in standard form and identify the conic section.0891

So, this is almost in the general standard form, but not quite.0894

I have 4x²; I need to move this -3y² next, then -16x - 18y - 12 = 0.0897

Now, I can easily see that a = 4 and c = -3; since these have opposite signs, that means that this is an equation describing a hyperbola.0907

OK, the next task is to complete the square.0921

I am going to first add 12 to both sides to remove the constant from the left side.0926

Then, I am going to group the x terms, which is 4x² - 16x.0937

And then, I have a -3y² - 18y, and that all equals 12.0946

I have a leading coefficient that is something other than 1, so I am going to factor out the 4, leaving behind x² - 4x.0955

Here, I need to factor out a -3; that is going to leave behind y² + 6y.0965

You need to be careful with the signs here; just double-checking: -3 times y² is -3y².0969

-3 times positive 6y is -18y, when you factor out with that negative sign; equals 12.0976

Completing the square: b²/4, in this case, is 4²/4, is 16/4; that is 4.0987

So, I am going to add 4 here; I am also going to add 4 times 4, or 16, to the right, to keep the equation balanced.0998

For the y expression, I have y² + 6y; therefore, b²/4 = 6²/4, which is 36/4, which is 9.1012

-3 times 9 is -27; so I am going to subtract 27 from the right side, again keeping the equation balanced.1029

I am rewriting this as (x - 2)² - 3(y + 3)² = 16 + 12, is 28, minus 27; conveniently, I end up with a 1 on the right.1041

Now, this is almost in standard form; generally, with standard form for a hyperbola, this term will be in the denominator.1059

So, it is possible to rewrite it like this; and it might be easier to look at it that way,1071

so that you can immediately know that this is a², instead of having to think it out.1077

Putting it in truly standard form is also a good idea, because recall that, if I have the numerator divided by 1/4, that is the same as this times 4.1082

And that tells me that I have a hyperbola with a center at (2,-3); you have to watch out for this positive sign.1096

And it has a horizontal transverse axis.1103

So, today we learned exactly what conic sections are, where they come from,1109

and how to look at an equation and determine what type of conic section it describes.1114

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