General Conic Equation and Eccentricity Examples

Which type of conic section is this?

2x² – 3y² – 4x + 2y – 12 = 0

Is this the good kind of conic, or the bad kind? Or some other kind entirely? Let's label all of our important constants to start off.

We only care about A and C, because the squared terms are the only ones that determine what type of conic we're dealing with.

A goes with x², and that's 2. C = -3. Neither of them is 0, so we don't have a circle or a parabola.

AC = -6, which, if we remember our number line, is less than 0. That means we have a hyperbola.

What is the eccentricity of an ellipse?

The eccentricity equals f, the distance to the foci, over a, the length of the semi-major axis. The foci of an ellipse are on the inside, so they have to be less than the semi-major axis: f < a. However, if f = 0, then we have a bit, fat, round circle. Putting that together, we get:

0 < f < a

Uh, we don't see the eccentricity anywhere in all of that. Let's divide all of this by a to get an e in the middle.

0 < e < 1

Much better.

What is the eccentricity of ?

To find the eccentricity, we only need f and a. Except, this isn't always true for a hyperbola (note: this is a hyperbola).

When talking about hyperbolas, we said that a is always associated with x. However, we don't want that for calculating the eccentricity.

Instead, we want the distance from the center to the vertices. For a vertical hyperbola (note: this problem has a vertical hyperbola), that distance is given by b. So, we'll use b = 3 for finding e.

a isn't getting off the hook, though, because we need it to find f.

f² = a² + b² = 9 + 16 = 25

f = 5

Okay, just one more plug-in of variables into an equation and we are done.