10. Analytical geometry

Analytic Geometry is a discipline that uses Algebra to study Geometry. In Enem, this subject is responsible for a significant portion of Mathematics questions, so it is essential that the student has a good understanding of the subject.

The study of Analytical Geometry begins with the introduction of the Cartesian plane, a tool that allows representing points and figures in two-dimensional space. The Cartesian plane is composed of two perpendicular lines, called axes, which intersect at a point called the origin. Each point on the plane can be represented by a pair of numbers, called coordinates, which indicate the distance from the point to each of the axes.

One of the main subjects studied in Analytical Geometry is the equation of the line. The general equation of the line is given by ax + by + c = 0, where a, b and c are constants and x and y are the coordinates of any point on the line. From this equation, it is possible to determine the slope of the line, the relative position of two lines and the distance between a point and a line.

Another important topic is the equation of the circle, which is given by (x - a)² + (y - b)² = r², where (a, b) are the coordinates of the center of the circle and r is the radius. From this equation, it is possible to determine the relative position of a circle and a line, the relative position of two circles, and the distance between a point and a circle.

Analytical Geometry also studies the ellipse, the parabola and the hyperbola, which are called conics. Conics are defined by quadratic equations and have interesting properties that are often explored in Enem questions.

The ellipse is defined by the equation (x/a)² + (y/b)² = 1, where a and b are the half-distances of the major and minor axes, respectively. The parabola is defined by the equation y = ax² + bx + c, where a, b and c are constants. The hyperbola is defined by the equation (x/a)² - (y/b)² = 1, where a and b are the half-distances of the transverse and conjugate axes, respectively.

In addition to these topics, Analytical Geometry also studies the relative position of points, lines and conics, the distance between points, lines and conics, and geometric transformations, such as translation, rotation and scale. These topics are fundamental to solve the most complex questions on the Enem.

Finally, it is important to remember that Analytical Geometry is a discipline that requires a lot of practice. Therefore, in addition to studying the theory, it is essential that the student solves many exercises and questions from previous tests. In this way, he will be able to develop the ability to apply theoretical concepts in solving practical problems, which is essential to obtain a good grade in the Enem.

In short, Analytical Geometry is a challenging discipline, but also a very interesting and useful one. With a good study and a lot of practice, the student will certainly be able to master this subject and obtain an excellent grade in the Mathematics test of the Enem.

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