7.5 Conic Sections

[STARTS AT 11:11]

Instructor: So put the equation 9x squared plus 4y squared minus 36x plus 24y plus 36 equals 0 into standard form and then graph the ellipse. 

So what we did with the parabola, we have to complete the square. We're going to do that twice. All right, so this will be 9x squared minus 36x plus something-- I'd like to know what that is-- plus 4y squared plus 24y plus something equals negative 36-- I'm just going to go and move that over-- plus something. 

All right, now I'm going to need to have just an x minus h or y minus k. So I'm going to go ahead and factor this out. X squared minus 4x plus something plus 4y squared plus 6y plus blank. So negative 36 plus that something. OK, so if I complete each of these squares, that is going to be a plus 4. Which means I really need to have a 36 here, 9 times that value. 

And then down here, that would make a plus 9, which means I really need to have a 36 there as well. So I'm going to add 72 to both sides there. So this will be 9. And then that is an x minus 2 squared plus 4y plus 3 squared equals 36. 

Now this needs to be equal to 1, so we're going to divide by 36. So I have x minus 2 squared over 4 plus y plus 3 squared over 9 equals 1. So this ellipse is centered at the point 2, negative 3, 2, negative 3, all right? 

Now, my major axis, my major axis is a value of 3 in the vertical direction. So my end points are here and here of my ellipse. My minor, my minor axis is 2. So going horizontally 2 in either direction, that gives me my ellipse, all right? 

Now, I really would like to know where the foci are. OK, well, the foci are going to be on either end. Now, what we need to do is find in this case our major axis is vertical. So we're looking at that second equation right there to find the foci, h,k plus or minus c. 

Now, we could find a value of c. C squared equals a squared minus b squared. Well, our e squared is 9. Let's go ahead and write that. 3 squared minus b squared-- whoop, switch that. No. Yeah, that should be good. 

9 minus 4, OK this is going to tell us the direction we need to go. Oh, that's 2. That's why that looks strange, 

OK, there we go. 9 minus 4, so that's 5. So our c is the square root of 5. So if we started our vertex and go along the major axis, square root of 5, which is 2 point some odd, our foci are going to be right around there. Again, starting with our vertex, going in opposite directions square root of 5. 

[ENDS AT 15:53]