Lesson 7 Conic Sections Identification

Conic sections

Formed by the intersection of a plane with a right circular cone.

Axis

Central line of symmetry around which the two nappes are generated.

Vertex

It’s the tip or sharp point where the two cones are joined together.

Vertex Angle

The angle formed at the vertex between the two sides (generators) of one nappe of the cone.

Generator

Passes through the vertex, and moves (or revolves) around the axis.

Directrix

Appears only when you cut the double-napped cone with a plane to form a conic section.

Eccentricity

It’s the constant ratio between how far a point is from the focus and from the directrix.

Circle

e = 0

Ellipse

0 < e < 1

Parabola

e = 1

Hyperbola

e > 1

Degenerate Conics

Special or limiting cases of regular conics. They show what happens when the cone and plane intersect in extreme positions.

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

General quadratic equation of a conic.

Circle

Plane cuts the cone perpendicular to its axis.

Ellipse

Plane cuts the cone obliquely (at an angle) but does not pass through the base.

Parabola

Plane is parallel to the slant edge of the cone.

Hyperbola

Plane cuts both nappes of the cone.

A point

Passes through vertex and perpendicular to axis.

A single line

Passes through vertex and parallel to a generator (slant edge)

Two intersecting lines

Passes through vertex and cuts both nappes

Parabola

Condition B² − 4AC = 0

Circle

B² − 4AC < 0, and A = C, B = 0

Ellipse

B² − 4AC < 0, and A ≠ C

Hyperbola

B² − 4AC > 0