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Conic Sections

Up to this point we've covered four different conic sections. We looked at circles then ellipses, hyperbola and finally parabola. We call each of these a conic section because they come from the intersection of a plane with a cone. So, here we have a cone and we have a plane that intersects it, creating this shape. Here's another cone that has a plane cutting straight across through it. And notice that this plane is horizontal to the base of the cone. For this shape, we have a plane cutting at an angle down to the side of the cone. And, finally, for this shape, We have a plane cutting through a double cone. So what I want you to do is match each of these conic sections with the picture that is represented by it. So if you think this shape is a circle, you would write a in this box. Good luck here.

Conic Sections

Here are the correct answers. Great job matching each one with its picture if you got them all right. We know that c is a circle since we'll have a perfect slice of our cone. And part of the reason that we create a circle is since this plane is parallel to the base of the cone. The radius from the center to any point on the blue circle will all be the same. For the ellipse, it's going to be b. Here we'll have a longer length this way than we will vertically this way. d will be our hyperbola since we see the two pieces that point away from one another and this leaves a to be our parabola. We just have one curved section that's cutting through the cone.

Equations of Conic Sections

So when we're determining conic sections, we want to look at the equation. We can tell by the equation whether or not the shape will be a circle, an elipse, a hyperbola, or a parabola. For the case of circles, they will always have the same amount of x squared and y squared. Meaning the numbers in front of x squared and the number in front of y squared will be the same. Even if I change these to 2 I still have a shape of a circle. And changing into 5 I still have a perfect circle. So let's go back to a perfect circle that has a radius of 10 units. We can change this equation to become an ellipse. All we need to do is make the coefficients of x squared and y squared different. For example, if I make this x squared term have a coefficient of four, instead of one, I get an ellipse. The key idea for circles and ellipses is that there's a positive sign in between the x squared term and the y squared term. The difference between them is that circles have the same coefficient whereas ellipses, have different coefficients. Here the coefficient of x squared is 4, and here the coefficient of y squared is 1. We usually just don't list the 1. But notice it's the same exact graph. Now if we want to turn this ellipse into a hyperbola. We'll want to change this addition sign to subtraction. Hyperbolas will always have a subtraction sign between their x squared term and their y squared term. So, changing this to a minus, we get our hyperbola. Remember, the minus sign is like the knife that cuts open our ellipse and inverts the sides so they point outward. And since our x squared term comes first our hyperbola opens to the left and to the right. They point in these directions. But let's see what happens when we make the x squared term negative, and the y squared term positive. We have subtraction between the squared terms, so we know this graph will end up being a hyperbola. And look, it is. The only difference is that now, our hyperbola opens up and it opens down. And that's because the y squared term comes first, it's positive. Whereas, the x squared term is negative. Now, there's one more conic to look at, the parabola. A parabola has x squared or y squared, but it doesn't have both. So in order to change this hyperbola into a parabola let's change y squared into just y. Now, I'm also going to change the value 100 to 10, so that way we can see the graph on here. And notice, when we change y squared to y, we get a parabola. The vertex is at zero negative 10, and the parabola opens upward. And here's how you can tell if a parabola will open up or if it'll open down. We can rearrange this equation and solve for y. So first I'll subtract negative 4x squared from both sides of this equation. This gives me negative y on the left, negative 4x squared on the right, and positive 10 on the right. Next I'll multiply every term by negative 1. This will give me 1y equals positive 4x squared minus 10. And we can easily see that all three of these equations represent the same graph. I could graph this one and I could graph this one and see that they all make the same parabola. Since this parabola is y equals we know its going to open up. Or it's going to open down. And since the coefficient on the squared term is positive, it opens up. Now, if we make this coefficient negative, the parabola opens down. So, when determining whether or not something opens up, or something opens down, you want to solve for the single variable first, and then look at the coefficient of the squared term. Is it positive or is it negative. So we've seen a parabola open up and down but let's see one that opens left and right instead of having the parabola be y equals let's make it be x equals so we'll make the x term be a single x variable. And the y term be y squared. Doing that we see we get a sideways parabola and it points to the right and again we know this parabola opens to the right since we can rearrange this equation to be x equals. First we'll add y squared to both sides so we'll get 4x equals positive y squared on the right. And positive 10 on the right. Next we'll divide every term by 4, to get x equals 1 fourth y squared plus 10 fourth. And when I graph each of these equations, I should see that they're all the same. Since this parabola is in the form x equals, we know it's going to open to the right or to the left. And since this number, this coefficient is positive 1 fourth. Our graph opens to the right. If instead we make it negative 1 fourth, our parabola will open to the left. So remember y equal parabolas open up and down. And x equal parabolas open right and left. The sign on the squared term tells us which direction that is. Up and right for positive. Down and left for negative.

Identify Conics 1

So now that we've reviewed all the shpaes that we've encountered so far, let's see if you can identify a couple of graphs. I want you to identify the type of conic this equation represents. Is it a circle, a hyperbola, an ellipse, or a parabola. Then if it's a hyperbola or a parabola, check all the directions in which it opens.

Identify Conics 1

This equation represents a parabola that opens up. Great work if you got that one right. If we divide each of these coefficients by 3, we'll get the equation y equals one third x squared, minus one third. Notice that the equation is y equals, and we have an x squared term. Since we only have an x squared term, and just a single y, we know it's a parabola. We don't have both x squared and y squared. Also, since this parabola starts with a y. And since it's in y equals, we know it's either going to open up or it's going to open down. We then look to the coefficient in front of x squared and see that it's positive. So hey, our parabola opens up.

Identify Conics 2

What do you think for this equation. Which of the conics would it be and if it's a hyperbola or parabola which direction does it open?

Identify Conics 2

This equation would just be an ellipse. Now you might have thought that it was hyperbola since we have a subtraction of y squared but remember we need the x squared and y squared to be on the same side of the equation. So if we add y squared to both sides of the equation we would get this one. We can see that there's a plus sign in between the x squared terms and the y squared terms so we either have a circle or no ellipse. And since we have different amounts of x squared and y squared we know that the equation represents an ellipse.

Identify Conics 3

What about this equation, which conic do you think it represents.

Identify Conics 3

This equation really represents a hyperbola. And it opens up and down. Nice thinking if you got that correct. Now remember, we want to get the x squared and y squared term on the same side of the equation. So, I'm going to subtract right, and 2y squared minus 2x squared on the left. Since we have a subtraction sign between the y squared and the x squared term, we know the conic is a hyperbola. And since this y squared term comes first, we know that our hyperbola opens up and it opens down. We can put this equation into the general form that we recognize, by dividing each term by 18. We'll have y squared divided by 9 minus x squared divided by 9 equals 1. And just to be sure, I can show you that each of these equations represent the same exact hyperbola. The equations are just written in different forms.

Identify Conics 4

How about this equation? What conic do you think it represents.

Identify Conics 4

This conic section would just be a circle. Great work if you chose that one. Just like before, we want to get the X squared term on the same side as the y squared term. So we can add x squared to both sides of the equation to get this equation. Now, you might not be used to seeing, the y squared term come first, and in fact, we can reverse the order of addition since it's commutative. Two plus one would be the same thing as one plus two. So y squared plus x squared is the same as x squared plus y squared. And just to be sure, I'll show you each of these equations graphed, so we see that they represent the same exact circle. This is the first one, this is the second one, and this is the third one. All three circles are exactly the same.

Identify Conics 5

Here's your fifth conic, what you think this one represents?

Identify Conics 5

This equation represents a hyperbola that opens to the left and to the right. Now we might not recognize the hyperbola in this form. So, let's subtract y squared from both sides of the equation. This'll move the y squared to the left-hand side, so we'll have x squared minus y squared equals positive 9 on the right. We have subtraction between our x squared term and our y squared term, so we know this is a hyperbola. Further more, we know that this parabola opens to the left and to the right since the x squared term is positive. It's the one that comes first. Now it's not so much which one comes first, but it's really the one that's positive determines the direction. Since the x squared term is positive, we know it opens left and right. And just to be sure, I'll show you this graph to tell you that these graphs are identical. The equations represent the same hyperbola.

Identify Conics 6

Here's another one. What do you think this equation represents? A circle? An ellipse? A hyperbola? Or a parabola?

Identify Conics 6

This equation represents an ellipse. We want to move this negative 4 y squared term to the left hand side of the equation. So we'll add 4 y squared here and we'll add 4 y squared here. So we'll be left with 27 on the right hand side. And 3 x squared plus 4 y squared on the left hand side. We have addition between x squared and y squared, so we know we have either a circle or an ellipse. And since, we have different amounts of x squared and y squared, we know we have an ellipse. And just to check in Desmos, I'll show you that this graph is exactly the same.

Identify Conics 7

Alright try finding out what this equation represents. Choose the best choice here and remember if its a hyperbola or a parabola tell me if it opens right or left up or down. Check all that apply.

Identify Conics 7

This equation represents a hyperbola that opens to the left and to the right. I hope you've got the hang of this. We'll start by moving 2 y squared to the left hand side. So we need to subtract 2 y squared here and subtract 2 y squared here. This leaves us with this equation. We have subtraction between our x squared term an our y squared term, so we definitely know that the conic, is a hyperbola. Now to determine which direction it opens, we look at which term is positive. Well the x squared term is positive. So that's why we know our hyperbola opens left, an it opens to the right.

Identify Conics 8

Alright how about this one. What shape do you think this equation represents.

Identify Conics 8

This equation will be a parabola that opens downward. Great work if you got it. We know it's a parabola since we have an x squared term, but just a y term. We don't have y squared. Because we don't have both x squared and y squared, we know our shape will be a parabola. To figure out which direction the parabola opens, we need to solve for y. So, first I'll subtract 27 from both sides of the equation. We subtract 27 here and here, to get this new equation. Now we want to divide every term by negative 4 to get y by itself. So, dividing this by negative 4, we get y. Dividing this by negative 4, we get positive 27 divided by 4 and dividing this by negative 4 we get negative three fourths x squared. So since this equation is of the form y equals, we know that it's going to open up, or it's going to open down. Since the x squared term has a negative coefficient, our parabola opens downward. And, just to show you, this equation is the one I have shown graphed, this equation is still the same. And this one is identical as well. All three equations represent the same parabola that opens down.

Identify Conics 9

And for our final conic section what do you think this equation represents. Choose the best choice and then if it's a hyperbola or a parabola. Let me know which way it opens.

Identify Conics 9

This equation represents a circle. Nice job if you picked the circle. We know we have equal amounts of x squared and y squared, and if we move the 3y squared to the left hand side by adding 3y squared to both sides of the equation, we'll get this equation. We have equal amounts of x squared and y squared, so we have a perfect circle. And notice, too, that we could divide each of these terms by plus y squared equals 9. This is a circle with a radius of 3 units. And just to be sure I'll show you that each of these graphs are exactly the same.