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Conic Section Formulas - Concept

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!

We can easily identify a conic section by its formula. Conic section formulas have different identifiers. For example, a vertical parabola has a squared "x" term and single "y" term while a horizontal parabola has a single "x" term and a "y" squared term. An equation for a circle has a squared "x" term, a squared "y" term and identical coefficients.

And now I want to take a couple minutes to just give you a general recap on conic sections and basically the very various equations that you've learned and key things that you can take from each equation okay? So what I have is written a variety of different equations and in general I just have them written in a way that you should hopefully see them or at least recognize them sometimes you're going to have to manipulate them like this first equation here this is standard form for a vertical parabola okay. You also sometimes see it in vertex form but hopefully you'll be able to switch back and forth between those two fairly easily.
So this first one what our graph is, is a vertical parabola like I just mentioned and basically what that means is it's going to be going up or down okay? It could be going facing upwards, it could be facing downwards it depends on that coefficient and key iding points are basically we have a y single y and a x squared okay? So those are the things you should look for in order to have a vertical parobola.
Next thing I want to talk about is the x is equal to y squared parabola, very similar but instead of going up and down this parabola is just going to go side to side and it could either be opening right or opening left again depending on that coefficient. Iding points for this single x, y squared okay.
Next conic is going to be a circle in this circle that I have here is actually moved over we still have our center being shifted but we know what a equation for a circle looks like and that is just going to be something like this and the key identifying pieces for this are, we have a x squared we have a y squared and they both have the same coefficients okay? Sometimes you'll see this equal to r squared also sometimes you can divide by r squared what would end up happening is we'd have these fractions that look sort of elliptical but if our x and radius x radius and y radius are the same that's why it's going to be a circle so what we idea is that is we have x squared y squared and the same coefficients okay some other conics that we've talked about.
The next one is an equation for ellipse again the center for this ellipse has moved but what we know with that graph looks like is going to basically be a oval okay? Given this particular example we don't know if it's going to be longer or taller that all depends on whether the a or b is bigger but we know that's an ellipse and how we can tell that is we have a x squared plus y squared and different coefficients okay? So next bit is almost identical in terms of the equation but instead of dealing with a plus we're dealing with a minus okay, the minus should be a key that we are dealing with a hyperbola the x term is telling you in this case that it's going to be going side to side so in this one we have something that looks like this, key identifying features in this case are x squared minus y squared.
The last little one we have is basically a hyperbola again we're dealing with a x squared and a y squared this time the y squared comes first this tells me that we have a vertical hyperbola key identifying features are y squared minus x squared okay.
A lot of different information up here but hopefully that sort of helped you weed out identifying which is which. Last little key points I want to talk about are some other relationships we have with specifically ellipses and hyperbolas and that is to find the focus okay? So to find the focus on a ellipse we can write that in writing here we end up with a squared minus b squared is equal to c squared. For a hyperbola the relationship with a squared and b squared are just addition instead of subtraction okay and the way that I remember these differences is basically we know we have a positive to make to identify a ellipse then we have a minus with our relationship with our focus we have a minus in our equation for hyperbola we then have a plus for our relationship with our focus so basically the signs are opposite you're either going to have one positive or one negative in each relationship.
So just a brief overview of all the different types of conics we've looked at.

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