Conic Sections

When a plane (also called as cutting plane) intersects with the nappes (one or both) of a double cone, a non-degenerated curve can be made and these curves are called conic sections. A conic section may take either of the following forms: a circle or an ellipse, a hyperbola or parabola, and even a point, a line or two intersecting lines in rare cases. A circle can be generated when the cutting plane is perpendicular to the axis of the cones. On the other hand, an ellipse or parabola can be generated when the cutting plane is not perpendicular to the axis and intersects only one nappe of the cone. A hyperbola can be generated when the cutting plane cuts through both the nappe of the cone. In rare cases wherein the cutting plane is perpendicular to the axis of the cone and intersects to the vertex of the cone, a point is generated. A line on the other hand is made when the cutting plane intersects the vertex and is parallel to a side on the cone. Still, an intersecting line can be generated when the cutting plane intersects both the nappes of the cone as well as the vertex.

A fixed point that relates the construction of the conic sections as well as its properties is known as the focus. A directrix is the line that serves to define the conic section as the set of all the points that satisfies a certain condition together with the focus. On the other hand the eccentricity of a conic section is the distance to the focus and the directrix. The eccentricity of a circle is equal to 0 while the eccentricity e of an ellipse is greater than 0 but lower than one. ON the other hand, a parabola has an eccentricity of 1 while e is greater than 1 for hyperbolas. Thus, through their eccentricity, you can determine what kind of conic section exists. Noncircular ellipse and as well as hyperbolas have two distinct foci associated with two distinct directrices and these directrixes are perpendicular to a certain line that connects the two foci.

The general equation of a conic section can be written as Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 however, every conic section has its own equations especially in computing for their vertices that helps in determining the kind of conic section given by a particular equation. The type of section can also be found from the equation B2 – 4ac. If B2 – 4ac is less than o then the conic is either an ellipse, a circle, a point or no curve while value of B2 – 4ac equal to 0 means that the conic is either a 2 parallel lines, 1 line, a parabola or no curve. On the other hand if the value of B2 – 4ac is greater then 0, the conic section is either 2 intersecting lines or a hyperbola.

For a circle, the horizontal and vertical vertex are equal and can be measured using the equation x2 + y2 = r2, where r is the radius of the circle and as mentioned earlier, the eccentricity of a circle is 0 and thus, every point on the curve have the same distance from the focus and in this case, the center of the circle.

For an ellipse, the horizontal vertex is given by the equation x2 / a2 + y2 / b2 = 1while the vertical vertex is given by y2 / a2 + x2 / b2 = 1 wherein: a is the radius of the ellipse or ½ of the length of the major axis, b is the minor radius and is ½ of the minor axis and c is the distance from the center to the focus of the ellipse. The distance c is also given by the equation a2 – b2 = c2 and the sum of the distances to each focus of the ellipse is constant.

For a given parabola, the horizontal vertex can be measured by the equation 4px = y2 while the vertical vertex is given by the equation 4py = x2 wherein p is the directrix or the distance from the vertex to the focus. The eccentricity of a parabola is equal to c/a and the distance to the focus is equal to the distance to the directrix.

For a hyperbola on the other hand, the horizontal vertex is given by the equation x2 / a2 – y2 / b2 = 1 and the vertical vertex is y2 / a2 – x2 / b2 = 1 wherein “a” is one half of the length of the major axis and “b” is one half the length of the minor axis and “c” is the distance of the center to the focus. For a horizontal hyperbola, the equation of the asymptote is y = ± (b/a)x while on a vertical hyperbola, the asymptote is given by the equation x = ± (b/a)y. The eccentricity of a hyperbola is equal to c/a and the difference between distances to each foci of the hyperbola is equal to a constant, a2 + b2 = c2.

For any conic with a center (j,k) rather than (0,0), each equation containing x and y should be replaced with (x-j) and (y-k) respectively. Thus for a circle instead of the equation x2 + y2 = r2, (x-j)2 + (y-k)2 = r2 should be used and for an ellipse, the equation x2 / a2 + y2 / b2 = 1 will be replaced by (x-j)2 / a2 + (y-k)2 / b2 = 1 and so on in measuring the vertex. The same goes in finding all the points in the given conics, just replaces all x’s with (x-j) and y’s with (y-k)

Most of the time, conics seemed to have no value in our everyday life that lessens our enjoyment and willingness to learn. However, conic sections have different uses in everyday life and we can see conics of different form, i.e. ellipse, hyperbola to mention a few, in our everyday life and in our surroundings. Conic sections also play an important part in many fields and in creating a better country, a better place to live in.

Indeed we can see ellipse and other conics almost everywhere even from a glass of water. There are also some establishments today that are elliptical in shape. Ellipse is also used to reflect sound wave and as well as light because of its property to reflect any signal or light that starts from one focus to the other. The principle is also being used in the field of medicine in the treatment of kidney stones. Parabolas on the other hand are use commonly as reflectors that could also include light and sound waves. It is also use in antennas and telescopes and as well as mirrors and eye glasses to correct the sight of a person. Hyperbolas on the other are also being used in the field of communication such as in airplanes. There are also different structures today that acquire the shape of a hyperbola such as steam power plants and other building. There also exists a Long Range Navigation system or Loran that uses the principle and characteristic of hyperbolas to synchronized signals especially radio signals.

Thus, conic sections even parabolas, ellipses and hyperbolas really do play a very important role in our development and even in our daily life. If only we are aware that such conics exist in our surrounding, we will have a better understanding, interest and appreciation of conic sections and their role in our society.

References

Britton, J. (04 January 2008). Occurrence of the Conics. Retrieved March 14, 2008 from

http://britton.disted.camosun.bc.ca/jbconics.htm.

Sellers, J. A. (No Date). An introduction to conic sections. Retrieved March 14, 2008 from

http://www.krellinst.org/UCES/archive/resources/conics/.

Wolfram Research Inc. (2008). Conic Section. Retrieved March 14, 2008 from

http://mathworld.wolfram.com/ConicSection.html.