DEFINITION Conic sections are plane curves that can be formed by cutting a double right circular cone with a plane at various angles.
AXIS DOUBLE RIGHT CIRCULAR CONE A circle is formed when the plane intersects one cone and is perpendicular to the axis
An ellipse is formed when the plane intersects one cone and is NOT perpendicular to the axis.
A parabola is formed when the plane intersects one cone and is parallel to the edge of the cone.
A hyperbola is formed when the plane intersects both cones.
In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. It can be defined as the locus of points whose distances are in a fixed ratio to some point, called a focus, and some line, called a directrix.
GENERAL EQUATION OF CONICS 𝑨𝒙𝟐+𝑩𝒙𝒚+𝑪𝒚𝟐+𝑫𝒙+𝑬𝒚+𝑭=𝟎
DISCRIMINANT Ellipse Parabola Hyperbola 𝑩𝟐−𝟒𝑨𝑪<𝟎
Parabola: A = 0 or C = 0 Circle: A = C Ellipse: A = B, but both have the same sign Hyperbola: A and C have Different signs
The Parabola The parabolais a set of points which are equidistant from a fixed point (the focus) and the fixed line (the directrix).
PROPERTIES The line through the focus perpendicular to the directrix is called the axis of symmetry or simply the axis of the curve. The point where the axis intersects the curve is the vertex of the parabola. The vertex (denoted by V) is a point midway between the focus and directrix.
The undirected distance from V to F is a positive number denoted by |a|.
The line through F perpendicular to the axis is called the latus rectum whose length is |4a|. The endpoints are 𝑳𝟏and𝑳𝟐. This determines how the wide the parabola opens.
The line parallel to the latus rectum is called the directrix.