Flash Cards: Conic Sections

A conic section is a curve obtained by intersecting a plane with a double-napped right circular cone.

The four types are circles, ellipses, parabolas, and hyperbolas.

(x – h)² + (y – k)² = r².

An ellipse is the set of all points in a plane where the sum of the distances from two fixed points (foci) is constant.

For a horizontal ellipse: (x²/a²) + (y²/b²) = 1, where a > b.

A parabola is a set of points equidistant from a fixed point (focus) and a fixed line (directrix).

y² = 4ax (opens right), y² = -4ax (opens left), x² = 4ay (opens up), x² = -4ay (opens down).

A hyperbola is the set of all points in a plane where the difference of the distances to two fixed points (foci) is constant.

For a horizontal hyperbola: (x²/a²) - (y²/b²) = 1.

The latus rectum is a line segment perpendicular to the axis of the parabola through the focus, with endpoints on the parabola.

The length is 4a.

a² = b² + c², where a is the semi-major axis, b is the semi-minor axis, and c is the distance from the center to a focus.

Eccentricity measures how much a conic section deviates from being circular, defined as e = c/a.

When the angle of intersection (β) between the plane and the cone is 90°.

An ellipse is characterized by the condition α < β < 90° for the plane's angle with the axis.

A parabola occurs when β = α, where the plane is inclined at the same angle as the generator of the cone.

A hyperbola forms when 0 ≤ β < α, cutting through both nappes of the cone.

Degenerate conics occur when the intersection is at the vertex of the cone, leading to points, lines, or pairs of intersecting lines.