INSTRUCTOR: We can graph conics that have been rotated. The conics, the standard forms that we've looked at all have a principal axis in the vertical or the horizontal directions. But suppose we wanted to look at something that had been rotated a bit. So I'm going to go and just do a quick little sketch here. And I'm going to go ahead and do a-- we'll do an ellipse to keep it simple. 

So here is an ellipse that has not been rotated. But suppose we wanted to rotate this. I can just take this and rotate it. Now, that's not to say that we're not still on a regular axes. Let's go ahead and just draw our normal axes. So there's a standard vertical y-axis and our horizontal axis. 

And so if we have an ellipse like this, we can see that with respect to these axes here, that equation would be pretty easy to find. However, with respect to a standard xy coordinate system, this would be kind of hard to find. 

And what we're going to do is we're going to label this here as the-- we're going to call this the x prime and y prime system, where the equation of this conic would be really easy to find in this system, whereas this original, this will be the x and y. And we'd like to find, what's the relationship between these? 

And then when we're given an equation of a conic in x and y that is not in our standard form, we can rewrite it in terms of x and y prime in a standard form that will not have a rotation with respect to those axes. And so then we can graph it. So I'm going to just go through a quick derivation here of what the relationship is going to be. 

What I'm going to do here is just pick a random point here. And it doesn't really matter what conic I'm looking at. But let's just suppose that I've got a point on the conic. Now, this point can be labeled both with respect to the x prime, y prime axes or with respect to the xy axes. So at this point, we'll call it the point x prime y prime, but we'll also call just the point x, y. And the question is, well, what is the relationship between those? 

So what we're going to do is actually just look at the line from the origin. And note that the origin is the same for both of these. And we can look at a couple of different triangles. I'm going to first look at this, connecting down to the x-axis with a right triangle. And then I'm going to connect it to the x prime axis with another right angle. 

And now I can see, OK, there's some angles here to consider. OK. This angle right here, I'm going to call this angle theta. And this angle here I'll call alpha. And then what we're going to do is-- oh, and also, I'm going to label this distance right here as r, the distance from the origin to our point. So alpha represents the angle of rotation, and theta represents the angle that our point makes with the x prime axis. 

Now I can just write some relationships using some trigonometric functions. I'll start with the x-axis or the xy system. x-- well, if I look at x-- sorry, x right here, and this triangle, which has a theta plus alpha angle, I can do the cosine. The cosine of this the sum of these two angles would be the x value over r. In other words, x is equal to r times cosine of theta plus alpha. 

OK. Similarly, I have y equals r times sine of theta plus alpha. So that's looking at this larger triangle here that comes down to the x-axis. But we can also get a relationship between x prime and y prime. x prime is going to equal r cosine of theta, cosine theta. And y prime will be r sine theta. 

So I've got all of these four different ways of locating this point, these two that are in the xy system and these two that are in the x prime, y prime system. And now I can use the sum formulas for cosine and sine to rewrite the x and the y. So I'm going to go do that over here. 

So x is r times-- the cosine of theta plus alpha is cosine theta, cosine alpha, and then minus-- still r, r-- and then sine-- sorry, sine theta, sine alpha. But I know that r cosine theta is x prime and I know that r sine theta is y prime. So then I get that this is x prime cosine theta minus y prime sine theta. 

And similarly, I can look at y. y is r sine of theta plus alpha. So again, using my sum identity there, I get r sine of theta cosine of alpha plus still r and then cosine theta sine alpha. And then here, r sine theta, that's y prime. So I have y prime cosine of alpha. And then plus r cosine theta is x prime sine of alpha. 

So these here, this is a relationship that exists between our xy values and our x prime, y prime values. You can do a very similar process to get a relationship that takes us from x to x prime and y prime. I'll just go ahead and write the formulas that we get. x prime is x cosine of alpha plus y sine of alpha. And y prime is y cosine of alpha minus x sine of alpha. 

So these are the formulas that relate the variables x prime, y prime and x and y. So anytime we want to rotate a conic, we can find this angle of rotation, alpha. And we could put them in here or up here. And then we can make replacements in an equation for a conic to switch between an xy system and an x prime, y prime system. 

Now, an important thing to know is that the standard equation of a conic, the most general that we can get, is going to look something like this, A x squared plus Bxy plus C y squared plus Dx plus Ey plus F equals 0. This is as general as we can get. You'll note that we have this xy term here. 

The presence of an xy term, meaning if B is not 0, tells us that we have a rotated conic. And so if we're given a form that's like this, we can then take these two, these expressions here, replace the x and the y with those, and this term will actually will disappear. And, oh, this should be an alpha. I'm sorry about that-- alpha. And the xy term will disappear. And we'll end up with an equation in the x prime, y prime system that we can actually graph.