7.5 Conic Sections

[STARTS AT 32:19]

Instructor: So identify the conic and calculate the angle of rotation described by this equation. Well, first we have a equals 13. B equals negative 6 root 3. I am going to write these in order. 

C is 7. D is actually 0. There is no x term. And e is also 0. There's no y term. And f is negative 256, OK. 

4ac minus b squared, if we calculate that, that is 413 times 7 minus b squared, negative 6 root 3 squared. That is equal to 256, which is greater than 0. That tells us we have an ellipse. So we are dealing with a rotated ellipse. 

Now, how much is rotated, that depends, all right. So let's find our angle of rotation cotangent 2 theta equals a minus c over b. So that is 13 minus 7 over negative 6 root 3, which is going to simplify to be negative 1 over root 3. Which is equal to negative root 3/3, OK. 

Now that is cotangent. So that implies that 2 theta is 2 pi over 3 and that theta is pi over 3. So we have a rotation of pi thirds, or 60 degrees. 

Now, let's use our values of a, b, and c to calculate the new coordinates. And now we have pi thirds there. We'll fill in this equation. Cosine squared of pi over 3 plus b, that's negative 6 root 3 times cosine of pi over 3 plus 7 sine squared of pi over 3, which is equal to 4. 

So our a value, our new a value in the rotated scheme is 4. B prime is just equal to 0. And that was based off of our equation. So it equals 0. All right, easy enough. 

Our c prime is our a value, that is 13 sine squared pi thirds minus b. So minus our negative 6 root 3, OK? Sine of pi over 3 times cosine of pi over 3 plus our c value of 7 cosine squared pi over 3, which is a total of 16. 

So our new c value is 16. Our d prime is going to be d. And our value of d was 0. Cosine theta, it doesn't actually matter what theta is in this case. Our value of e was also 0. So our d prime is 0. For the exact same reason, e prime is 0 because our d and e values are 0. And f prime is going to be equal to f, which is negative 256. 

This brings our new equation to 4x squared plus 16y squared minus 256 equals 0. If we rearrange this, ad 256 to both sides, and then divide by 256, we get x squared over 64 plus y squared over 16 equals 1. And this is rotated by pi over 3. 

So our center is 0,0, a major axis of 8. But it's rotated pi over 3. So it's going to go in this direction, 8 in this direction, 8, and then 4 going in this direction, 4 going in this direction. And we have our ellipse roughly. 

OK, that brings us to the end of this section and also to the end of this chapter. Be sure that you understand what's going on here. So all of this stuff is really building. 

We started with some geometry, the focal parameter, the eccentricity, all the different ways to define what these conic sections are. But we want to end up in polar coordinates. That's kind of how these tie together. And parametric equations lead into that. This is all to prepare you for what might come next.

[ENDS AT 38:34]