Conic Sections

Conic sections are plane curves resulting from the intersection of a cone's surface with a plane. Their study traces back to ancient times; Menaechmus is credited with their discovery in the 4th century BC. They are fundamental to the field of geometry and appear in many applications, e.g. in engineering, physics, architecture, design, computer graphics, and astronomy. Historically most conic sections were known as conics, but in contemporary usage the singular rarely appears without the plural sections. 

Types of conic sections

There are five types of conic sections: the circle, ellipse, parabola, hyperbola, and degenerate conic. The circle is the locus of all points in a plane equidistant from a given point called the center. The ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. The parabola is a plane curve with a vertex and a directrix line; for every point on the curve, the distance between the point and the directrix is equal to the distance between the point and the vertex. The hyperbola is a plane curve that consists of two connected components, each of which is asymptotic to a pair of intersecting lines called asymptotes. A degenerate conic is a conic section formed by the intersection of the plane and the cone that results in a point, line, or pair of intersecting lines.

Equations of conic sections

Conic sections can be represented by equations in the Cartesian coordinate system. The general equation of a conic section is:

Ax² + Bxy + Cy² + Dx + Ey + F = 0

where A, B, C, D, E, and F are real numbers and A, B, and C are not all zero.