Key Points

Conic sections are curves formed by the intersection of a plane with a double-napped right circular cone. The primary types of conic sections are the circle, parabola, ellipse, and hyperbola.

A circle is the set of all points in a plane that are at a fixed distance (radius, $r$) from a fixed point (center, $(h, k)$). Its standard equation is $(x-h)^2 + (y-k)^2 = r^2$.

A parabola is the set of all points in a plane equidistant from a fixed point (focus) and a fixed line (directrix). The line passing through the focus and perpendicular to the directrix is the axis of the parabola.

For a parabola with vertex at the origin, the standard equations are $y^2 = 4ax$ (opens right), $y^2 = -4ax$ (opens left), $x^2 = 4ay$ (opens up), and $x^2 = -4ay$ (opens down).

The latus rectum is a line segment perpendicular to the axis of the parabola, passing through the focus, with endpoints on the parabola. Its length is $4a$ for all standard parabolas.

An ellipse is the set of all points in a plane where the sum of the distances from two fixed points (the foci) is a constant. This constant sum equals the length of the major axis ($2a$).

For an ellipse centered at the origin, the equations are $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (major axis on x-axis) or $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ (major axis on y-axis), with $a > b$.

In an ellipse, $a$ is the semi-major axis, $b$ is the semi-minor axis, and $c$ is the distance from the center to a focus. These are related by the equation $c^2 = a^2 - b^2$.

The eccentricity ($e$) of an ellipse measures its elongation and is defined as the ratio $e = \frac{c}{a}$. For an ellipse, $0 < e < 1$. A value of $e$ closer to 0 means the ellipse is more circular.

The latus rectum of an ellipse is a chord through a focus, perpendicular to the major axis. Its length is given by the formula $\frac{2b^2}{a}$.

A hyperbola is the set of all points in a plane where the absolute difference of the distances from two fixed points (the foci) is a constant. This constant difference equals the length of the transverse axis ($2a$).

For a hyperbola centered at the origin, the equations are $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (transverse axis on x-axis) or $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ (transverse axis on y-axis).

In a hyperbola, $a$ is the semi-transverse axis, $b$ is the semi-conjugate axis, and $c$ is the distance from the center to a focus. They are related by the equation $c^2 = a^2 + b^2$.

The eccentricity ($e$) of a hyperbola is defined as the ratio $e = \frac{c}{a}$. For any hyperbola, the eccentricity is always greater than 1, i.e., $e > 1$.

The latus rectum of a hyperbola is a chord through a focus, perpendicular to the transverse axis. Its length is given by the formula $\frac{2b^2}{a}$.

Let $\alpha$ be the semi-vertical angle of the cone and $\beta$ be the angle the intersecting plane makes with the cone's axis. A circle forms if $\beta = 90^\circ$; an ellipse if $\alpha < \beta < 90^\circ$; a parabola if $\beta = \alpha$; and a hyperbola if $0 \le \beta < \alpha$.