WEBVTT mathematics/pre-calculus/selhorst-jones
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Welcome to Educator.com.
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In the past four lectures, we have discussed various conic sections: parabolas, circles, ellipses, and hyperbolas.
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And this lecture is designed to bring that information together and to give you some context about this.
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First of all, what are conic sections? We know we can name them; we know what they are; but where do they come from?
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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),
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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.
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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.
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If you tip that plane at an angle, the result is an ellipse.
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If you encompass the edge of the come, you end up with a parabola.
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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.
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So, this is where conic sections come from; and they have many applications in science.
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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).
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This standard form that I am talking about now is a very general form.
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It gives you a general equation, ax² + bxy + cy² + dx + ey + f = 0.
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So, what we are going to do in a minute is talk about how you can look at this general form and determine
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which type of conic section you are working with, so that you can put the equation
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in the standard form particular to that type of conic section.
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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,
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what type of conic section you are working with; and now I am going to bring that all together.
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OK, if b = 0, we can analyze that standard form of the conic section to determine what type of conic section the equation represents.
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Looking back at that general standard form again, ax² + bxy + cy² + dx + ey + f = 0.
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Here, we are having the limitation that b = 0.
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And throughout this course, when we work with conic sections, we have only worked with ones where b is 0.
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When b is 0, you end up with this.
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Once you have this standard form, then you can go ahead and analyze it in ways we are going to discuss in a minute
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to determine which type of conic section you have (what the equation describes).
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But let's talk for a minute about what the xy tells you.
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So far, we have worked with shapes such as parabolas; and some were oriented vertically; some, horizontally.
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We also worked with ellipses (some had a horizontal major axis; some had a vertical major axis); and the same with hyperbolas.
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So, even though the center may have been shifted, these were all either strictly vertical or strictly horizontal.
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What this bxy term does is rotates it so that instead of, say, having an ellipse that has
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a completely vertical major axis or horizontal major axis, you could end up with an ellipse like this--
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the major axis is at an angle--or a parabola that is like this.
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And that is definitely more complicated to work with; and it doesn't allow us to complete the square, then,
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to shift an equation from the general form to a specific standard form.
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So, later on, if you continue on in math, you may end up working with these shapes.
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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.
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In order to identify conic sections, you need to look at the coefficients of the x² and y² terms.
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So, let's rewrite this; and again, the assumption is that b = 0.
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So, I am just going to have ax² + cy² + dx + ey + f = 0.
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Parabola: Recall that, with a parabola, you have an x² term or a y² term, but not both.
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Therefore, either a is 0 (so this drops out) or c is 0 (so this drops out).
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An example would be something like x = 3y² + 2y + 6.
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Or you might have y = 2x² - 4x + 8.
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So, neither of these has both an x² and a y² term in the same equation.
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For a circle, recall that what you are going to end up with is an x² and a y² term
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on the same side of the equation, with the same sign; and they are going to have the same coefficients.
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Therefore, a is going to equal c.
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An example would be x² + y² + 3x - 5y - 10 = 0.
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Here, a equals 1, and c equals 1; those are the same coefficients; x² and y²
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have the same sign and the same side of the equation; so it is a circle.
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If we are working with an ellipse, this time the x² term and y² terms are going to be
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on the same side of the equation, with the same sign (like with the circle), but a and c are different.
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They are unequal; that tells me that I am working with an ellipse.
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For example, 12x² + 9y² + 25x + 28y + 40 = 0.
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Here, I have a = 12 and c = 9; so this is the equation describing an ellipse.
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Finally, with a hyperbola, these are pretty straightforward to recognize, because you are going to have
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an x² term and a y² term, but they are going to have opposite signs.
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Their coefficients will have opposite signs.
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For example, 4x² - 8y² + 10x + 6y - 34 = 0.
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So, I have a = 4 and c = -8; since a and c have opposite signs, this is an equation describing a hyperbola.
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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."
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It is standard form, but it is a very general standard form for any type of conic section.
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OK, now we are going to work on identifying the various conic sections by looking at their equations.
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First, write in standard form, and identify the conic section.
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OK, so general standard form is what I am talking about right now: it is x² + 2y².
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I need to subtract 4x from both sides, subtract 12, and set everything equal to 0.
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What this tells me is that I have a = 1 and c = 2.
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Since a = 1 and c = 2, these have the same sign (the x² and the y² terms); but they have different coefficients.
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And that means that what I am working with is an ellipse.
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You could go on, then, and write this in the specific standard form for an ellipse.
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Let's do that by completing the square: start out by grouping...let's rewrite it here.
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And then, let's group the x and the y terms; so x² terms group together; y terms group together.
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And now, add 12 to both sides to move that over, to make completing the square a little bit easier.
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To complete the square for x² - 4x, I need to add b²/4.
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b²/4 is equal to 4²/4, is 16/4; it is 4.
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So, I add x² - 4x + 4; and it is very important to remember to add the 4 to the right side, as well.
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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.
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All right, that is x² - 4x + 4 + 2y² = 16.
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This can be rewritten as (x - 2)² + 2y² = 16.
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But recall, in standard form for an ellipse, you need to have a 1 on the right.
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So, rewrite this up here, and then divide both sides by 16.
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This is just (x - 2)²/16; this will cancel; this will become 8; and then 16/16 is 1.
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So, we started out with this equation, put it into the general standard form to identify that this is an ellipse,
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and then went on to complete the square; and now I have it written in standard form specifically for an ellipse,
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which is much more useful when you are working with that and trying to graph.
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This time, without completing the square, all we are going to do is identify the conic section.
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And this is already in standard form; therefore, a = 2; c = -3.
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Since a and c have opposite signs, this is the equation for a hyperbola.
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I have an x² term and a y² term, both, so it is not a parabola.
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They have opposite signs; therefore, it must describe a hyperbola.
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OK, write in standard form and identify the conic section.
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Right now, this is not in any type of standard form; so I am going to work with the general standard form.
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First, I am going to subtract 36x² from both sides.
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Then, I am going to subtract 128 from both sides.
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This means that I have a = -36, and c = 16; since these two are opposite signs, this is an equation describing a hyperbola.
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OK, now, let's go ahead and put this in standard form specific to a hyperbola.
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And let's start out by moving this 128 back over to the right; this is actually 32.
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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.
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That is -9x² + 4y² + 8y =...128/4 would give 32.
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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:
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4y² + 8y - 9x², because I am going to have a difference.
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To complete the square, I first need to factor out that 4; then I need to add b²/4 to this expression.
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This is going to equal 2²/4; that is 4/4, which is 1.
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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.
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At this point, I am going to rewrite this as (y + 1)² - 9x² = 36.
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The last step is: I want the right side to be 1, so I am going to divide both sides by 36.
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4 goes into 36 nine times; 9 goes into 36 four times; and then this cancels out to 1.
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OK, so I started out with an equation that wasn't in any kind of standard form.
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I put it in general standard form, and then determined it was a hyperbola, completed the square, and ended up
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with an equation in the standard form for a hyperbola, so that I can use that to graph the hyperbola, if needed.
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Write in standard form and identify the conic section.
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So, this is almost in the general standard form, but not quite.
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I have 4x²; I need to move this -3y² next, then -16x - 18y - 12 = 0.
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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.
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OK, the next task is to complete the square.
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I am going to first add 12 to both sides to remove the constant from the left side.
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Then, I am going to group the x terms, which is 4x² - 16x.
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And then, I have a -3y² - 18y, and that all equals 12.
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I have a leading coefficient that is something other than 1, so I am going to factor out the 4, leaving behind x² - 4x.
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Here, I need to factor out a -3; that is going to leave behind y² + 6y.
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You need to be careful with the signs here; just double-checking: -3 times y² is -3y².
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-3 times positive 6y is -18y, when you factor out with that negative sign; equals 12.
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Completing the square: b²/4, in this case, is 4²/4, is 16/4; that is 4.
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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.
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For the y expression, I have y² + 6y; therefore, b²/4 = 6²/4, which is 36/4, which is 9.
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-3 times 9 is -27; so I am going to subtract 27 from the right side, again keeping the equation balanced.
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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.
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Now, this is almost in standard form; generally, with standard form for a hyperbola, this term will be in the denominator.
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So, it is possible to rewrite it like this; and it might be easier to look at it that way,
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so that you can immediately know that this is a², instead of having to think it out.
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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.
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And that tells me that I have a hyperbola with a center at (2,-3); you have to watch out for this positive sign.
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And it has a horizontal transverse axis.
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So, today we learned exactly what conic sections are, where they come from,
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and how to look at an equation and determine what type of conic section it describes.
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