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Conic Section Formulas - Problem 1

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Whenever we are identifying conics the main thing we are concerned with is our relationship between x and y namely are we adding and subtracting and also the coefficients on terms if we are dealing with x²'s and y²'s.

So what I have behind me is 3 different conics and I’m not so much concerned with what these look like or where the center is or what the shape is anything like that, what I just want to do is look at them and say what kind of conic are they going to be.

So I’m just going to talk you through my logic I’m not actually going to write much in the board but hopefully if through just sort of hearing my talk process you will be able to understand sort of how we can easily figure this out.

So my first example what I see right away is I have a one term squared and a another term that is not. So that’s going to be the sign to me that we have a parabola. Every other conic we deal with circles, ellipses, hyperbolas all have both terms squared.

So here we know we have a y singular and an x² hopefully realize if we add these 3 over we have y is equal to x² plus something which basically is the equation for a parabola. So this I know is a parabola I also know it's facing upwards because our coefficient of x is positive and y is singular.

Next one, we have x²'s and y²'s so what I want to do is whenever we have x² and y² we want to get everything to one side. And in order to that what we have to do is subtract the y² over. So what I end up with is 4x² minus our y² is equal to a number. If we're going to solve this out we have to divide by this number but really what key thing I see is we have x² minus y². That should kill you in that we are looking at a hyperbola whenever we are dealing with a minus that means it’s a parabola and our x² term is first, which tells us we are dealing with a side to side hyperbola. We are going to be ending up dealing with something like this.

This one is actually centered at the origin so this graph isn’t exactly right but the main point was identifying it as a horizontal hyperbola. The last one again I see x² and y² so I know that I want to get everything over to the same side adding over this 7y². What I see then is that we have both coefficients are the same, our coefficient on x² and y² are 7. If we are to solve this out we would need to divide by 63 because we need it to be equal to 1 but no matter what happens if we have the same coefficients and we divide by a number then the coefficients are still the same.

Which tells me we are going to end up with x² plus y² is equal to something it’s going to be a circle. This one there’s no transformations because it’s not plus or minus so it's centered on the origin.

So just by thinking about how the equation is going to shape up, knowing what we know about conics with pluses and the minus and the squares we are able to fairly easily at least identify what kind of shape it is. Like I said we don’t know exactly what these graphs look like but we can get a rough idea if the shape we are going to be dealing with if we were to graph it.

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