Conic Section

• Conic Section is defined as a set of curves formed from dividing or cutting a right circular cone of two nappes with a plane.
• The cones are also called as nappes
• Generator lines meet at a single point called vertex
• Generator lines are lines that meet at a certain point
• Vertex is the point where two nappes intersect
• The axis of the cone is the line passing through the vertex and perpendicular to its nappes
• The surface base is a circular-shaped opening of the cones
• If the plane cuts only one of the circular cone nappes and it is parallel or facing the surface base of the right circular cone or it cuts all generators, the formed image is either circle or ellipse
• If the cutting plane is parallel only to one generator and it is perpendicular to the base of the cone, the curved formed is a parabola
• If the plane cuts both nappes and it is parallel to two generators the intersection curved is a hyperbola
• Axis - line passing through the vertex
• Generator - a line rotates about the vertex
• Vertex is the intersection of the generator and the axis of the cone
• Vertex Angle - an angle between the axis and generator
• Nappe - an lateral surface of right circular cone
• Directrix - the perimeter of the circular base
• A conic section (or simply conic) is the intersection of a plane and a double-napped cone.
• If the plane is perpendicular to the axis of revolution, the conic section is a CIRCLE.
• If the plane intersects one nappe at an angle to the axis (other than 90 degrees), then the conic section is ELLIPSE.
• If the plane is parallel to the generating line, the conic section is PARABOLA.
• If the plane is parallel to the axis of revolution (y-axis), then the conic section is a HYPERBOLA.
• A degenerate conic is generated when a plane

  intersects the vertex of the cone.
• The degenerate form of a circle or an ellipse is a singular point.
• The degenerate form of a parabola is a line or two parallel lines
• The degenerate form of hyperbola is two intersecting lines.
• if both x² and y² are present and A=C (Ax²+Bxy+Cy²+Dx+Ey+F=0) the conic is a circle
• if both x² and y² are present and A≠C (Ax²+Bxy+Cy²+Dx+Ey+F=0) but A and C have the same sign then the conic is an ellipse
• if both x² and y² are present and A≠C (Ax²+Bxy+Cy²+Dx+Ey+F=0) but A and C have the opposite sign then the conic is a hyperbola
• if either x² or y² are present then the conic is a parabola