Conic Sections
Circle as a Conic Section
A CIRCLE is the set of points in a plane that are equidistant from one point. That one point is called the CENTER of the circle, and the distance from it to any point on the circle is called the RADIUS of the circle. The standard form for the equation of a circle with its center at (0, 0) and with a radius of length 'r' is represented by the equation: ➜ x² + y² = r² The circle is the simplest and best known conic section. As a conic section, the circle is the intersection of a plane perpendicular to the cone's axis. The geometric definition of a circle is the locus of all points a constant distance from a point and forming the circumference (C). Copy and paste the following link into your browser to learn more about a circle as a conic section: http://www.bing.com/videos/search?q=Circle+as+a+Conic+Section&&view=detail&mid=3AF5E775044F40A4B4F63AF5E775044F40A4B4F6&FORM=VRDGAR
Parabola as a Conic Section
A PARABOLA is the set of points in a plane that are the same distance from a given point and a given line in that plane. The given point is called the FOCUS, and the line is called the DIRECTRIX. The midpoint of the perpendicular segment from the focus to the directrix is called the VERTEX OF THE PARABOLA. The line that passes through the vertex and focus is called the AXIS OF SYMMETRY. The equation of a parabola can be written in two basic forms: Form 1: y = a( x - h)² + k Form 2: x = a( y - k)² + h In Form 1, the parabola opens VERTICALLY, meaning that it opens in the "y" direction. If a > 0, it opens UPWARD. If a < 0, it opens DOWNWARD. The distance from the vertex to the focus and from the vertex to the directrix line are the same. This distance is ⎮1/4a⎮ A parabola with its vertex at (h, k), opening vertically, will have the following properties: ➜ The focus will be at (h, k + 1/4a). ➜ The directrix will have the following equation: y = k - 1/4a. ➜ The axis of symmetry will have the following equation x = h. ➜ Its form will be y = a(x - h)² + k. In Form 2, the parabola opens HORIZONTALLY, meaning it opens in the " x" direction. If a > 0, it opens to the RIGHT. If a < 0, it opens to the LEFT. A parabola with its vertex at (h, k), opening horizontally, will have the following properties. A parabola with its vertex at ( h, k), opening vertically, will have the following properties: ➜ The focus will be at (h + 1/4a, k). ➜ The directrix will have the following equation: x = h - 1/4a. ➜ The axis of symmetry will have the following equation y = k. ➜ Its form will be x = a( y - k)² + h. Copy and paste the following link into your browser to learn more about a parabola as a conic section: http://www.bing.com/videos/search?q=Parabola+as+a+Conic+Section&&view=detail&mid=6EF3B84DEB152FC364A06EF3B84DEB152FC364A0&FORM=VRDGAR
Ellipse as a Conic Section
An ELLIPSE is the set of points in a plane such that the sum of the distances from two fixed points in that plane stays constant. The two points are each called a FOCUS. The plural of focus is foci. The midpoint of the segment joining the foci is called the CENTER OF THE ELLIPSE. An ellipse has two axes of symmetry. The longer one is called the MAJOR AXIS, and the shorter one is called the MINOR AXIS. The two axes intersect at the center of the ellipse. The equation of an ellipse that is centered at (0, 0) and has its major axis along the x‐axis has the following standard form: ➜ (x² / a²) + (y² / b²) = 1, where a² > b² ▶︎▶︎The length of the major axis is 2|a|, and the length of the minor axis is 2|b|. ▶︎▶︎The endpoints of the major axis are (a, 0) and (-a, 0) and are referred to as the MAJOR INTERCEPTS. ▶︎▶︎The endpoints of the minor axis are (0, b) and (0, -b) and are referred to as the MINOR INTERCEPTS. ▶︎▶︎ (c, 0) and (-c, 0) are the locations of the foci, then c can be found using the equation ➜ c² = a² - b² If an ellipse has its major axis along the y‐axis and is centered at (0, 0), the standard form becomes: ➜ (x² / b²) + (y² / a²) = 1, where a² > b²² The endpoints of the major axis become (0, a) and (0, -a). The endpoints of the minor axis become ( b, 0) and (-b, 0). The foci are at (0, c) and (0, -c), with ➜ c² = a² - b² When an ellipse is written in standard form, the major axis direction is determined by noting which variable has the larger denominator. The major axis either lies along that variable's axis or is parallel to that variable's axis. Copy and paste the following link into your browser to learn more about an ellipse as a conic section: https://youtu.be/9xSgKqTxZbg
The Four Conic Sections
CONIC SECTIONS are formed on a plane when that plane slices through the edge of one or both of a pair of right circular cones stacked tip to tip. Whether the result is a CIRCLE, ELLIPSE, PARABOLA, or HYPERBOLA depends only upon the angle at which the plane slices through. Conic sections are described mathematically by quadratic equations—some of which contain more than one variable. When the edge of a single or stacked pair of right circular cones is sliced by a plane, the curved cross section formed by the plane and cone is called a CONIC SECTION. The four main conic sections are the circle, the parabola, the ellipse, and the hyperbola.
Hyperbola as a Conic Section
HYPERBOLA is the set of all points in a plane such that the absolute value of the difference of the distances between two fixed points stays constant. The two given points are the foci of the hyperbola, and the midpoint of the segment joining the foci is the center of the hyperbola. The hyperbola looks like two opposing "U‐shaped" curves. A hyperbola has two axes of symmetry. The axis along the direction the hyperbola opens is called the transverse axis. The conjugate axis passes through the center of the hyperbola and is perpendicular to the transverse axis. The points of intersection of the hyperbola and the transverse axis are called the VERTICES (singular, vertex) of the hyperbola. A hyperbola centered at (0, 0) whose transverse axis is along the x‐axis has the following equation as its standard form: ➜ (x² / a²) - (y² / b²) = 1 where (a, 0) and (-a, 0) are the vertices and (c, 0) and (-c, 0) are its foci. In the hyperbola, c² = a² + b. As points on a hyperbola get farther from its center, they get closer and closer to two lines called ASYMPTOTE LINES. The asymptote lines are used as guidelines in sketching the graph of a hyperbola. To graph the asymptote lines, form a rectangle by using the points (-a, b), (-a, -b), ( a, b), and ( a, -b) and draw its diagonals as extended lines. For the hyperbola centered at (0, 0) whose transverse axis is along the x‐axis, the equation of the asymptote lines becomes: equation ➜ y = ± (b / a)x Copy and paste the following link into your browser to learn more about a hyperbola as a conic section: https://youtu.be/qzPj0aQIUTU
