- All right. Well, if we're trying to identify conics given the algebraic equations that actually generate the particular curve, well, look down at this. OK. If we see this, it looks so familiar that we can see that it's an ellipse. But what if something is a little bit less obvious? Well, it turns out-- for example, let's suppose that someone gives us this. [LAUGHS] 

[GROANS] 

Well, what are you going to do here? Well, this is so complicated that it's not exactly in the nice form. We just can't immediately see what it is. Well, it turns out that if we think and just look about it, the coefficients in some sense will give us the clues. So let me actually show you a way of finding what's called the discriminant that will allow us to actually identify much more generic objects. 

So let's take a look at this little chart here. Again, I'm just showing you the most generic quadratic that I can think of in x and y. So it's got x, x squareds. It's got x times y's. It's got y squared, it's got x's. It's got y's. It's got constants. And I have the whole thing equal to 0. Notice that I have the expression written in this order something x squared plus something xy plus something y squared plus something x plus something y plus a constant equals 0. 

And I'm assuming that the A, the B, and the C don't all equal 0. Otherwise, if they all did, this would be a line. We could see exactly what it is. That would be a line, straight line. All right. Well, then we actually create what's called the discriminant. And the discriminant is exactly equal to B squared minus 4AC. 

And that actually is familiar. If you think back to the famous quadratic formula that we've seen from time to time, under the square root, under that radical, we actually see B squared minus 4AC. So it's the exact same quantity. Although, now, you'll notice that this is for A, this is for B, and this is for C. Anyway, if you compute that value and it turns out that that is negative and B equals 0 and A and C are equal, then, in fact, this identifies a circle. 

On the other hand, if we're negative, if the discriminant is negative, and either this is not true or this is not true, then we don't have a perfectly symmetric thing. We have just a mere ellipse. So if we have the discriminant being negative, then we know we have either a circle or an ellipse. If the discriminant is positive, then we just know immediately, we have a hyperbola. And if the discriminant is exactly 0, then in fact, it's a parabola. 

So it's pretty easy to identify just by looking at the coefficients. Now let's bring back our contestant. And you see that we can figure this out pretty easily because all we have to do is compute B squared minus AC. So let's do that. If we look at B squared minus AC, we see B squared, that's going to be negative 2 squared, which is 4. Let me write that out though. Negative 2 squared minus 4 times AC. 

And so what do I see? Well, I see this is going to be a 4, and this is going to be a minus 4 times 25. You can actually figure this out, right? This is going to be 100 and so forth. But I don't care. Look at it. You can instantly see this is much bigger than that. This number is less than 0. That's all I care about. Now, what is it? Well, these two quantities are equal. So A equals C. That's great. But notice that B is not 0. So since B is not 0, that means if we look back at our table, I have this value being negative, but B is not 0. So therefore, I'm an ellipse. 

So this is really an ellipse in disguise, an ellipse in disguise. Wow. So you can identify them, even when they look really exotic. Let's take a look at another one, just to warm up. Warming up. Let's figure out B squared minus AC here. So B squared is going to be 7 squared, which is 49, minus 4 times A, 2, C, which is negative 3. 

So here, I see 49. And then I see 8. And 8 times negative 3 is actually negative 24, but I've got a minus sign in front. So that becomes a positive 24. And so that's some number. But all I care about is that it is positive. That's clear. It's positive. And when you look at this chart, you see that if, in fact, that quantity is positive, we're a hyperbola. We're a hyperbola. That's right. We're a hyperbola-- hyperbola. 

And you've got it. So you can identify a conic section if you're given a very general quadratic in x and y by just looking at the coefficients carefully and computing the discriminant. Cool.