Conics

CIRCLES

Conics

1Conic sections are plane curves that can be formed by cutting a double right circular cone with a plane at various angles. DEFINITIONDOUBLE RIGHT CIRCULAR CONEA circle is formed when the plane intersects one cone and is perpendicular to the axis

AXISAn 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.

5A hyperbola is formed when the plane intersects both cones.

DEGENERATE CONICInanalytic geometry, a conic may be defined as aplane algebraic curveof degree 2. It can be defined as thelocusof points whose distances are in a fixed ratio to some point, called afocus, and some line, called adirectrix.GENERAL EQUATION OF CONICSEllipseParabolaHyperbolaDISCRIMINANTParabola: A = 0 or C = 0Circle: A = CEllipse:A = B, but both have the same signHyperbola:A and C have Different signsThe parabola is a set of points which are equidistant from a fixed point (the focus) and the fixed line (the directrix).The ParabolaPROPERTIESThe 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.Axis of SymmetryDirectrixLatus Rectum \4a\VertexFocus|a|TYPES OF PARABOLA

TYPE 1

TYPE 2

TYPE 3

TYPE 4Sample Problemsolution

| | | | | 1 2 3 4 5 | | | | | -5 -4 -3 -2 -1 | | | 1 2 3 | | | -3 -2 -1 Sketch the graphs and determine the coordinates of V, F, ends of LR, and equation of the directrix.