"The three types of conic section are the hyperbola, the parabola, and the ellipse."
Understanding parabolas, ellipses, and hyperbolas.
Cartesian Coordinates: This is the system of coordinates used to plot graphs in the plane, it involves plotting points in a plane using an x,y coordinate system.
Distance Formula: This is a formula that is used to find the distance between two points on a graph.
Midpoint Formula: This formula is used to find the midpoint of a line segment.
Equations of Lines: Different forms of equations are used for Lines such as the slope-intercept forms or point slope form equations.
Parabola: The Parabola is a conic section that is obtained when a plane intersects the cone at an angle parallel to one of its sides.
Vertex: The vertex is the point on the Parabola that is closest to the focus, which is the point at which the Parabola curves.
Directrix: The directrix is a line perpendicular to the axis of symmetry that intersects the Parabola.
Axis of Symmetry: The axis of symmetry is a line that passes through the vertex and divides the Parabola into two equal halves.
Standard Form: The standard form of the equation of a Parabola is y = a(x-h)^2 + k, where the vertex is (h,k).
Circle: A circle is another type of conic section that is formed when a plane intersects a cone at a right angle to its axis.
Center: The center of a circle is the point that is equidistant from all points on the circle.
Radius: The radius of a circle is the distance from the center to any point on the circle.
Diameter: The diameter of a circle is the line segment that passes through the center and has endpoints on the circle.
Standard Form – The standard form of the equation of a circle is (x: H)^2 + (y - k)^2 = r^2, where (h,k) is the center of the circle, and r is the radius.
Ellipse: An ellipse is a conic section formed when a plane intersects a cone at an angle that is not parallel or perpendicular to its axis.
Major Axis: The major axis of an ellipse is the longest diameter of the ellipse.
Minor Axis: The minor axis of an ellipse is the perpendicular line segment that intersects the major axis at the center of the ellipse.
Foci: There are two foci in an ellipse, which are the points on the major axis that determine the shape of the ellipse.
Standard Form: The standard form of the equation of an ellipse is (x-h)^2/a^2 + (y-k)^2/b^2 = 1 or (x-h)^2/b^2 + (y-k)^2/a^2 = 1, where (h,k) is the center of the ellipse, a is the length of the major axis, and b is the length of the minor axis.
Hyperbola: A hyperbola is a conic section formed when a plane intersects a cone at an angle that is steeper than that required to form a Parabola but not steep enough to form two separate intersecting lines.
Center: The center of a hyperbola is the point that is equidistant from the foci.
Foci: There are two foci in a hyperbola, which determine the shape of the hyperbola.
Asymptotes: The asymptotes are two straight lines that intersect at the center of the hyperbola and approach the branches of the hyperbola as they extend across the plane.
Eccentricity: The eccentricity of a hyperbola is a measure of how far apart the foci are from the center.
Standard Form – The standard form of the equation of a hyperbola is (x-h)^2/a^2: Y-k)^2/b^2 = 1 or (y-k)^2/b^2 -(x-h)^2/a^2 = 1, where (h,k) is the center of the hyperbola, a is the length of the distance from the center to a vertex, and b is the length of the distance from the center to the asymptotes.
Circle: A circle is a set of all points on a plane that are at a fixed distance from a center point. The distance from the center point to any point on the circle is the same.
Ellipse: An ellipse is a set of all points on a plane such that the sum of distances from any point on the ellipse to two fixed points is constant. The fixed points are the foci of the ellipse.
Parabola: A parabola is a set of all points on a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
Hyperbola: A hyperbola is a set of all points on a plane such that the difference of distances from any point on the hyperbola to two fixed points (the foci) is constant.
Degenerate Conics: A degenerate conic is a special case of a conic section where the cross-section becomes a point, line, or pair of intersecting lines. For example, a circle with a radius of zero is a point, and a parabola that intersects its focus is a line.
Non-real Conics: These are imaginary conic sections that do not have any real solutions. Examples of these are a hyperbola with two real roots that are both negative, or an ellipse that does not intersect the x or y-axis.
Rectangular Hyperbola: Also known as the equilateral hyperbola, this is a special type of hyperbola that is formed when the distance between the foci is equal to the distance from the center to the vertices.
Point Ellipse: A point ellipse is an ellipse with one of its foci and its center point coinciding.
Latus Rectum: The latus rectum is a line segment that passes through the focus of a parabola and is parallel to its axis of symmetry.
Eccentricity: This is a measure of the shape of a conic section, which is defined as the ratio of the distance between the foci to the length of the major axis. It ranges between 0 and 1, with a value of 0 representing a circle and a value of 1 representing a line.
"The circle is a special case of the ellipse, though it was sometimes called as a fourth type."
"The ancient Greek mathematicians studied conic sections, culminating around 200 BC."
"Apollonius of Perga's systematic work on their properties."
"A non-circular conic can be defined as the set of those points whose distances to some particular point, called a focus, and some particular line, called a directrix, are in a fixed ratio, called the eccentricity."
"The type of conic is determined by the value of the eccentricity."
"A conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables."
"The geometric properties of the conic can be deduced from its equation."
"By extending the Euclidean plane to include a line at infinity, obtaining a projective plane, the apparent difference vanishes."
"The branches of a hyperbola meet in two points at infinity, making it a single closed curve."
"The two ends of a parabola meet to make it a closed curve tangent to the line at infinity."
"Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically."
"The eccentricity is the fixed ratio in defining a conic section."
"A non-circular conic is the set of those points whose distances to some particular point and some particular line are in a fixed ratio."
"A conic may be defined as a plane algebraic curve of degree 2... satisfy a quadratic equation in two variables."
"The ancient Greek mathematicians studied conic sections, culminating around 200 BC."
"Apollonius of Perga's systematic work on their properties."
"The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions."
"The circle is a special case of the ellipse."
"The geometric properties of the conic can be deduced from its equation."