Suppose I have an ellipse/hyperbola rotated about the origin by some angle $\theta$. Am I right in saying that the following general process will find the eccentricity $e$ of these conics?
- Find $a$, the length of the semi-major axis. For ellipse, the major axis is the longest diameter, for the hyperbola, the major axis is the chord between the vertices.
- Find $b$, the length of the semi-minor axis.
- Then, $b^2 = a^2(1-e^2)$ for an ellipse, and $b^2 = a^2(e^2 - 1)$ for a hyperbola, allowing you to solve for the eccentricity $e$.
Then, in general, the focii are the two points that lie a distance $ae$ away from the origin, along the major axis.
Further, in general, the directrices are the lines parallel to the minor axis, at a distance $\frac{a}{e}$ away from the origin.