7.5 Conic Sections

[STARTS AT 17:56]

Instructor: Put the equation 9x squared minus 16y squared plus 36x plus 32y minus 124 equals 0 into standard form, and graph the resulting hyperbola. And then we want to find the asymptotes. All right, we're going to complete the square again. So 9x squared plus 36x plus something yet to be determined minus 16y squared plus 32y minus 124. Actually, let's move the 124 to the other side. 

Now I'm going to also, like I did previously, factor adding 9. Squared plus 4x plus blank minus 16 minus squared minus 2y plus something equals 124 plus something there. Now, completing the square in the first box should be a plus 4. 

So that means I actually want a plus 36 up here, since I did just divide it by 9 going down and multiply by 9 going up. Down here, that would be a plus 1, which means we have a minus 16 we included in there, which means in total we're adding 20 to both sides, all right. 

Now, factoring that we have x plus 2 squared minus 16. That will be y minus 1 squared equals 144. Dividing both sides by 144, that would be x plus 2 squared over 16 minus y minus 1 squared over 9 equals 1, OK? 

So I can tell that my center is negative 2,1. There is my center. Now, my vertices, well, my major axis is the x-axis. And I'm going 4 in either direction. OK, so let my vertex go 4 to the left and the right. My minor access, major is the x-axis. Just label that, Ooh, all right, my major and my minor, all right. 

So my value of a is 4. My value of b is 3. My value of h is negative 2. My value of k is 1. It's from my center. All right, now the minor axis is going 3 in either direction. So 1, 2, 3. All right, so I'm going to go ahead and sketch this box, really, because I want you to see where I'm going with this, all right. 

So I have my two vertices. And if I were to draw the diagonals of this box, that rectangle, if I were to draw these, those are my two asymptotes. My function is going to go through the vertex and approach those 2 asymptotes. And that is what it's going to look like. 

Now, I should probably emphasize here that these asymptotes are not generally going to make them dashed. They're generally not drawn as solid lines because well, we get close to those, they're not actually part of our graph itself. 

Now, the equations of those are y equals plus or minus, or k plus or minus b over a times x minus h because we have a major axis that is horizontal. So we have 1 plus or minus b divided by a. That would be 3/4x minus h. So that's plus 2. And there are our two oblique asymptotes for this. 

[ENDS AT 23:01]