The Ellipse - Problem 4
Finding information about a conic section when it's not in a form we recognize. So what we're looking at right now is this long string of terms and we're trying to figure out what exactly this is. I'll tell you right now this is a ellipse, but we can go through the process to figure out how I know that.
The first thing we want to do when we see something like this is to change it into a form we know and how we do that is by completing the square. First thing we want to do when we're completing the square is get all of our variable that relate near each other and pull out any other constant terms to the other side.
So what I'm going to do is write all my x's together 4x² plus 24x, write all my y's together plus y² minus 4y and then pull this 36 around equals -36.
So from here when we complete the square, we have to have a coefficient of 1. So looking at my 4x² what I see is I don't have that so I need to factor out a 4 strictly from my x terms. My coefficient on y² is 1 which is perfectly fine, so I can leave that as it is. So what we end up with here is then 4x² plus 6x leave the space for when we complete the square plus y² minus 4y again leaving a space for when we complete the square equals -36. So what we need to do is complete the square.
We have our 4 out in front divide our middle term by 2 so that's going to be 3 and whatever that term that we just added in squared is what we add inside of our parenthesis. So we added 9 inside of our parenthesis, but what we have to be careful about is we're actually multiplying that by 4, so we actually added 36 to this term. To keep it balanced we have to add 36 to the other side.
Completing the square for the other term, we don't have a coefficient this time, dividing our -4 by 2 will give us -2, squaring that we'll end up with the +4, we need to add the 4 to the other side as well and I think I left off my little squares here didn't I? There's a square and there's a square and this is then going to be equal to -36 plus 36 those cancel plus 4, so I end up with equal to 4.
Whenever we're dealing with ellipses, we have a positive sign in between which we do here, but we're always to 1, so in order to get this into a form we know, we need to divide by 4, so this divide by 4 ending up with x plus 3², our 4s cancel plus y minus 2² over 4 is equal to 1.
So finishing this up, what we are asked for is, well hopefully first your axis here this is an ellipse, we're dealing with a plus sign and it's equal to 1, so therefore we know it's an ellipse. So extracting the information we know, this is x plus 3 it's telling us our center is back 3 units y minus 2 is telling us our center is up 2, so our center then, where should I write this we'll write it over here? is going to be equal to -3, 2.
Next thing we're looking for is the length of the major axis. The major axis is the longer one no matter what the direction is and what we look at there is the denominators our fraction. Here are long this is always a term squared, so here we're saying our radius in the y direction is 2. Here we're saying our radius in the x direction is actually 1. Remember that it's over nothing, it's actually over one so this tells us our longer radius is 2, our radius in one direction, our radius in the other direction, so actually our diameter, our major axis is 4. So major is going to be 4 units long and our minor then.
Our minor axis is going to end up being this term squared, actually our minor radius, so our minor radius is 1 unit. So our minor diameter is that times 2, multiply 1 by 2 and we end up with 2 as our minor axis. All I need you to remember that your axis is the entire length.
So completing the square in order to figure out what this conic section is and then extracting the information from the equation that we know.