Math Exam Syllabus PDF

# Conic Sections ## Circumference The circumference is a geometric figure that consists of the set of all points in a plane that are at a constant distance (called radius) from a fixed point called the center. ### Characteristics: * **Center:** The point from which all the points of the circumfe...

# Conic Sections ## Circumference The circumference is a geometric figure that consists of the set of all points in a plane that are at a constant distance (called radius) from a fixed point called the center. ### Characteristics: * **Center:** The point from which all the points of the circumference are at the same distance. * **Radius:** The distance between the center and any point of the circumference. * **Diameter:** The segment that joins two points of the circumference passing through the center, and its length is double the radius. * **Chord:** A segment that connects two points of the circumference but does not pass through the center. * **Arc:** Part of the circumference limited by two points. The equation of a circumference on a Cartesian plane with center at (h, k) and radius r is: $(x - h)^2 + (y - k)^2 = r^2$ ## Hyperbola The hyperbola is an open curve that is formed when a plane cuts a cone at an angle such that it does not intersect the base of the cone. It is defined as the set of points whose difference in distances to two fixed points (foci) is constant. ### Characteristics: * **Axes:** There are two axes in a hyperbola, the major axis and the minor axis. * **Foci:** Two fixed points that are fundamental for the definition of the hyperbola. * **Asymptotes:** Lines that approach the hyperbola but never touch it. The standard equation of a hyperbola centered at the origin is: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ## Parabola The parabola is a symmetrical curve that is formed when a plane cuts a cone parallel to one of its sides. It can be defined as the set of points that are equidistant from a fixed point (focus) and a fixed line (directrix). ### Characteristics: * **Vertex:** The point closest to the focus, where the parabola changes direction. * **Focus:** The fixed point from which the distance is measured. * **Directrix:** The fixed line used to define the parabola. The standard equation of a parabola with vertex at the origin that opens upwards is: $y = ax^2$ ## Ellipse The ellipse is a closed figure that results from cutting a cone with a plane at an angle such that it intersects both parts of the cone. It is defined as the set of points whose sum of distances to two fixed points (foci) is constant. ### Characteristics: * **Major and minor axes:** The length of the major axis is greater than that of the minor axis. * **Foci:** Two fixed points inside the ellipse. The standard equation of an ellipse centered at the origin is: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where a is the semi-major axis and b is the semi-minor axis. These concepts are fundamental in analytic geometry and have applications in various fields such as physics, astronomy, and graphic design.