The five conic sections are the circle, ellipse, parabola, hyperbola, and degenerate conics (point, line, and two intersecting lines).
A conic can be defined using a general second-degree equation, such as Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. The coefficients A, B, C, D, E, and F determine the type and orientation of the conic.
The focus-directrix property states that for any point on a conic, the distance to a fixed point (the focus) is equal to the distance to a fixed line (the directrix). This property is used to define and distinguish different types of conics.
Conics have various applications in fields such as engineering, physics, and astronomy. For example, parabolas are used in satellite dish antennas, ellipses are used in the orbits of planets, and hyperbolas are used in wireless communication.
The eccentricity of a conic is found using the formula e = √(1 - (b^2/a^2)), where a and b are the lengths of the semi-major and semi-minor axes, respectively. The eccentricity determines the shape of the conic and can range from 0 (circle) to 1 (parabola) to values greater than 1 (hyperbola).