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When $A$ and $C$ are both not zero and have same signs, then the conic will be an ellipse. For example, $x^{2}+2 y^{2}=1 .$When $A$ and $C$ are both not zero and have different signs, then the conic will be a hyperbola. For example, $x^{2}-2 y^{2}=1 .$When $A$ or $C$ are is zero, then the conic will be a parabola. For example, $x^{2}=2 y .$

Calculus 2 / BC

Algebra

Chapter 8

Analytic Geometry in Two and Three Dimensions

Section 4

Translation and Rotation of Axes

Polar Coordinates

Introduction to Conic Section

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So in this problem, we're going to discuss how do the charming when the general question of iconic section in this case a X Square plus B x y Lucy y Square was the X plus y plus F is equal to zero, he said. Lips, hyperbole or a problem by just looking at this is general form. So for this type of questions, we're gonna assume that A and C cannot be zero the same time. A can be zero but see cannot in vice versa. So for any leaps, if a and see have same signs, that means they're both positive or they're both negative. The journal form of thesis equation is just gonna leave, too. He's just gonna lead you on the lips. An example of thes is X Square plus two Y Square is equal to one. It doesn't matter how. What's the value of a in what's the value of C? As long as they both have the same sign they're going to jail than the lips. If they're both equal, that's just a special case. If a sickle to see this is just a circle, no for hyperbole. If A and C have different signs that means positive, the negative or negative in positive. This will be a parable. I hate parabolas. Sorry and an example. It's X squared, minus two y square tickle toe one. You can see right here that this is going to be our hyperba, that it's gonna be like playing in the X axis. And by the negative sign, you could determine that this will be a hair hey problem and for a problem. If a sequel to zero, let's see it's not or vice versa. Then this will be a problem. So it doesn't matter the sign off a or C that whatever it's not sirrah. It will be a problem, and it's simple. It's excess square. Sequel to two wine in just a society note it will be. This will be a problem opening towards the first degree trim in this case, the term that has a first degree for his degree. That means ex or just why it will be the one that determines where it's the problem opening towards. So in this case, this parable over here, X squared is equal to two y. It's opening towards the positive. Why axe the busted of Y axis, because why is the long term in this case? And it depends on next, not on the other the other way around.

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