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Conic section from expanded equation: circle & parabola

Sal manipulates the equation x^2+y^2-3x+4y=4 in order to find that it represents a circle, and the equation 2x^2+y+12x+16=0 in order to find it represents a parabola. Created by Sal Khan.
Video transcript
Let's see if we could do a couple more of these conic section identification problems. So I have this problem, x squared plus y squared minus 2x plus 4y is equal to 4. And so, the first thing I like to do is just try to figure out what type of conic section this will be. And we have our -- this is my x squared term, my y squared term. They're on the same side of the equation. And they both have positive coefficients. So this tells me that we're going to be dealing with an ellipse. And in this case in particular, the coefficients are the same number. They're both positive 1. So this is going to tell me that this is a circle. So let's get this in the standard form. And try to graph the circle. So we're going to want to complete the square. So let's take the x terms, since we get x squared minus 2x plus something to complete the square later on. Plus, and now let's do the y squared terms. y squared plus 4y plus something is equal to 4. And now what are we at here? We take 1/2 of the minus 2, minus 1. Square it. That becomes plus 1. Add a 1. We have nothing out here, so we really just added a 1 to the left-hand side of this equation. So we have to add 1 to the right-hand side. And here we take 1/2 of 4. 1/2 of 4 is 2. 2 squared is 4. Put a 4 here. So you have to add a 4 to the right-hand side as well. And we actually did add just a 4, because there was nothing multiplying the 4 out here. And so this becomes x minus 1 squared plus y plus 2 squared is equal to 4 plus 1 plus 4 is equal to 9. And there you have it. We have it in the standard form of a circle. You remember that if a circle is centered at 0, the standard form would be x squared plus y squared is equal to r squared. So this is r squared, this is the radius squared. So that tells us the radius of the circle is 3. And it was just shifted so that its origin, instead of being 0, 0, is at the point 1, minus 2. And the reason why we got 1, minus 2, we just have to think about what makes this whole expression equal 0? In this case it was the origin. In this case, it's x equals 1. And what makes this whole expression equal 0, in this case, it was y is equal to 0. In this case it's y is equal to minus 2. So that's our center. That's our radius, and we're ready to graph this circle. So it's at, let me see. I should graph the circle first. That's fair enough. So it's going to be at 1, minus 2. So, 1, minus 2. So it's going to be down here. So it's going to come out, this circle's going to start there. It's going to be at 1, and then you go 1, 2. So that's pretty close to the center. Maybe I should do it at 1, 2. So that's at center. Right there, at 1, minus 2. And then its radius is 3. So this distance right here is 3, in any direction. 3. And that is 3. Fair enough, that was a pretty straightforward problem. Circles in some ways are the simplest. And remember I said it's going to be an ellipse. And you say, oh, this isn't the standard formula for an ellipse. So just as a refresher, if you divide both sides of this equation by 9, what do you get? You get x minus 1 squared over 9 plus y plus 2 squared over 9 is equal to 1. And then you see that the horizontal axis, I guess. Or the horizontal diameter, is going to be 3. Or the horizontal radius is going to be 3. And the vertical radius is also going to be 3. Because the radius never changes in this ellipse. Which is really a circle. Let's do one more. Just to make sure you know this stuff cold. So I have 2x squared plus y plus 12x plus 16 is equal to 0. Let's look at the x squared and the y squared terms. 0's an squared term. But I don't see a y squared term. So this is a bit of a conundrum. And this will lead us to the fourth of our conic sections, which I talk about in the first video but we haven't really touched on yet. And that's the parabola. And how do I know it's a parabola? You're familiar, and I'll go more in future videos on all the different ways that a parabola comes about. And how all the points are equidistant between one point and a line and all of that. But just in very simple ways, you recognize the most simple parabola is y is equal to x squared. That parabola looks something like this. Where its minimum point, or its vertex, is at the origin. Or if you have a parabola like, x is equal to y squared. That looks something like this. Where it's a sideways version of that one. Where, once again, its vertex is at the origin. And just out of interest, so we know that this is a parabola, because we have a y and we have an x squared, right? They're different degrees. There's no second degree term of the y. And just to put this in a form that's familiar to you, let's just subtract everything but the y from the left-hand side. So you get y is equal to minus 2x squared minus 12x minus 16. And this is kind of the traditional form that you're familiar with. You're probably even used to finding the 0'z of this parabola, and we could do that right now. We could say, OK, when does this equation intersect the x axis. The x axis is when y is equal to 0. So that equals 0. You get minus 2x squared minus 12x minus 16. And this is different to what we normally do. Normally I would immediately break into completing the square. But I just want to figure out the 0's of this parabola first. So this 0's equal to minus 2 times factoring out minus 2, you get x plus 6x plus 8. So the 0 is equal to minus 2 times x plus 2 times x plus 4, I just factored that. And so for this whole thing to be 0, either this is 0 or that is 0. And so, either x plus 2 is equal to 0, or x plus 4 is equal to 0. So x is equal to minus 2, and x is equal to minus 4. That's the two 0's of this parabola. So we immediately know one thing about this parabola, and you've probably already done this in your algebra classes. If we were to draw the x axis, it intersects the x axis at 1 and minus 2, and 3 and minus 4. That's all we know about this right now. So let's see if we can use some of our completing the square skills with the conic sections we've done so far, to to come up with a little bit more information about this parabola. So let's try to complete the square with it. I'll rewrite it down here. So it's y is equal to, this is the one I'm dealing with. Now, let me just take the x terms by themselves, and factor out the minus 2. Minus 2 times x squared plus 6x. And I'm going to add something else. And then I have a minus 16 over there. To make this a perfect square, I have to take 1/2 of this, 1/2 of the 6. 1/2 of the 6 is 3. 3 squared is 9. If I add nine to the right-hand side of the equation, remember, I didn't just add 9. This is 9 times minus 2 I'm adding. So if I subtract, that's minus 18, if I subtract 18 from the right-hand side, I also have to do it from the left-hand side. So, subtract 18 there. And now my equation becomes y minus 18 is equal to minus 2 times, what is this? This is x plus 3 squared minus 16. And let's just get it in a form that we might start recognizing from our conic sections. Let's add 16 to both sides. If we add 16 to both sides, y minus 18, plus 16. It's going to be y minus 2, I'll put parentheses around that. Is equal to minus 2 times x plus 3 squared. And you might wonder why I put it in this form. And I did because this'll help us, this is kind of the same pattern that you see with all of the other conic sections. Like, if I were to tell you to graph y is equal to x squared, y is equal to x squared. It would look something like, let me draw the - whoops - Draw some axes here. y equals x squared looks something like this. It looks like a parabola. I mean, it is a parabola, with the vertex at 0, or its minimum point. And that's what the vertex is, is the minimum point of, or the maximum point of the parabola. We'll talk more about that, you'll learn a lot more about that when we go into calculus. But I think you can recognize it, the bottom of the U or the top of the U. If I were to draw y is equal to minus x squared, you could plot some points. But it looks something like that. If I were to try to graph y is equal to 2x squared, it would just be like the y is equal to x squared, but it would go up twice as fast. It would look something like that. So the vertex at the origin. And finally, if I were to graph y is equal to minus 2x squared, it would look something like this. It would open downward and it would go down twice as fast. Now, this one, this equation that we ended up with, right here, this is the same thing as y is equal to minus 2x squared. It has the same general shape. But instead of its vertex, or its center or its starting point, or whatever you want to call it, to be at the origin, it's shifted now. You say what y value makes this term zero, well, y is equal to 2. Just why, in this case, what y value makes this 0? Well, it's 0 here because we were at the origin. And here, what x value makes this 0? It's x is equal to minus 3. So this gives us information about where the vertex is. It's at x is equal to minus 3, y is equal to 2. So it's at x is equal to 1, 2, 3. y is equal to 1, 2. Right there. We already know those two points, because we've figure out it's 0. But even if we didn't know those two points -- we know that this has the same general shape as y is equal to minus 2x squared. So it's going to open downward. Like this. And a little bit faster than y is equal to minus x squared. So it's going to look like this. And we know it goes through that point and we know it goes through that point. There you go. So we've touched on every conic section. And in the next two videos, I'll go into a little bit more depth of the theory behind conic sections and how they came about and all of that. But I think now you're ready to tackle a lot of what you might actually see on your algebra test. See you soon.