46. ​​Conics

Conics are an important topic in the study of mathematics and are often asked in the Enem exam. The term 'conic' is derived from the fact that these curves can be obtained as intersections of a cone with a plane. There are three types of conics: the ellipse, the parabola and the hyperbola.

The parabola is the curve formed by the intersection of a cone with a plane parallel to its generatrix. In mathematical terms, a parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). The parabola has an important property that is used in many practical applications: light rays entering parallel to the axis of a parabola will reflect through the focus, and this property is used in car headlights and satellite dishes.

An elllipse is the curve formed by the intersection of a cone with an inclined plane at an angle less than the angle of the apex of the cone. An ellipse is the set of all points in a plane whose sum of distances from two fixed points (the foci) is constant. The ellipse has two lines of symmetry, which are the major axis and the minor axis. The orbits of the planets around the sun are ellipses, with the sun at one focus.

A hyperbola is the curve formed by the intersection of a cone with an inclined plane at an angle greater than the angle of the apex of the cone. A hyperbola is the set of all points in a plane whose difference in distances from two fixed points (the foci) is constant. The hyperbola has two branches opening in opposite directions and two lines of symmetry, which are the asymptotes of the hyperbola.

To better understand conics, it is important to study their equations. The general quadratic equation represents a conic. This equation is of the form Ax² + Bxy + Cy² + Dx + Ey + F = 0. Depending on the values ​​of the coefficients A, B and C, the equation can represent a parabola (if B² - 4AC = 0), an ellipse (if B² - 4AC < 0 and A = C) or a hyperbole (if B² - 4AC > 0).

Conics have many practical applications. As mentioned earlier, the reflective property of the parabola is used in car headlights and satellite dishes. Ellipse properties are used in physics to describe the orbits of planets and in engineering to draw arches for bridges and tunnels. Hyperbola properties are used in hyperbolics to represent the trajectory of subatomic particles in a magnetic field, and in engineering to draw the structure of certain types of buildings and bridges.

In summary, conics are an important part of the study of mathematics and have many practical applications. To prepare for the Enem test, it is important to understand the properties of conics, know how to obtain their equations and be familiar with their applications.

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