Submitted to:

Mr N.C Sharma

Submitted By:Harshika R.No 1Jyoti R.No 9CLASS XI - A

GENERAL

OVERVIEW

View of the Conic Sections

Historical View of the Conic Sections

In this hypertext, we consider the conic sections, which have beenstudied for over 2000 years. Many people have contributed to this study,and many historical references and texts exist to document this study. Apollonius of Perga, one of the greatest Greek mathematicians of the time(circa 200 B.C.), appears to have been the first to have rigorously studiedthe conic sections. He applied his work to his study of planetary motionand used this to aid in the development of Greek astronomy. (Recall that Perga was one of the cities visited by the apostle Paul duringhis first missionary journey as recorded in Acts 13:13. Paul would havebeen in Perga less than 300 years after Apollonius' development of thetopic of conic sections.) More information on Apollonius, as well as many other mathematicians, is held atthe MacTutor History of Mathematics Archive at the University of St. Andrews. Youmay view pages dealing with Apollonius, Hypatia, Euclid and many other peoplewho played important roles in the development of conics.

There are even Biblical references to such geometric objects. I Kings 7:23 discusses Solomon's "sea", which was to be a circular basin used for ceremonial cleansing. The verse reads as follows:

Now he made the sea of cast metal ten cubits from brim to brim, circular in form, and its height was five cubits, and thirty cubits in circumference.

New American Standard Bible

As a sidenote, observe that the value of is being estimated here, in some sense, by the integer 3. That is to say,

or

where C is the circumference and d is the diameter of a circle.

To close this section, I have included three images below which are reproductions of works held in the Vatican Exhibit Main Hall. This site contains many old manuscripts of writings in the areas of mathematics and astronomy, as well as concise background material to shed light on the historical significance of the documents.

Fuller descriptions of the three images below can be obtained at the Vatican Exhibit site. Moreover, the images below are black and white versions of the color originals held in the museum. If you click on the titles of the texts from which these images are taken, you can see the color originals as held in the collection. Note that these require quite a bit of time to download, so do this only when convenient for you. I strongly encourage you to view these at some point, as the beauty of these documents can only be appreciated fully after one sees the color images.

The first of the images is a page from a Latin translation of Euclid's Optics written in 1458.

Secondly, here is a page from a Greek version (circa 1536) of Apollonius' Conics.

Finally, an elaborately developed page from a Latin translation of

Ptolemy's Almagest.

Algebraic View of the Conic SectionsWe now move from a geometric point of view to an algebraic approach. Note that all of the conics that we will see below will satisfy a second-degree equation of the form

Ax2 + Bxy + Cx2 + Dx + Ey + F = 0

where A, B, C, D, E, and F are constants. As we change the values of some of these constants, the shape of the corresponding conic will also change. Hence, it will be very important to focus on these differences in the algebraic equations as we study the individual conic sections. Knowing the subtle differences in the equations will help us to quickly identify the type of conic that is represented by a given equation

x2 + y2 – 25 = 0is of the form mentioned above. It can be rewritten as

x2 + y2 = 25which you may recognize as the equation for a circle centered at the origin

with radius 5. So you see, you (probably) have worked with these algebraic equations

before. They will continue to appear as you move into the sections below dealing with the particular conic sections.

Geometric Origin of the Conic Sections

In this section, we are interested in the classical construction of the conic sections. Note the following diagram: INSERT DIAGRAM OF PLANE CUTTING RIGHT--CIRCULAR CONE It was Apollonius (mentioned above) who was the first to note that the conic sections could be constructed apart from algebraic equations by cutting the right--circular cone with a plane. As a matter of fact, Apollonius did not note the connection of the conics to their algebraic equations. These equations did not enter the mathematical picture for hundreds of years. Let us examine the figure above a little more closely. Note that if the right--circular cone is cut by a plane which is perpendicular to the axis of the cone, then the resulting intersection is a circle. (Note that the axis of the cone in the figure is simply the vertical line which goes through the vertex of the cone, the point at which the two pieces of the cone meet.) If the cutting plane is parallel to the ``side'' of one of the cones, then we see that a parabola is constructed. Thirdly, if the plane intersects both pieces of the cone, then a hyperbola appears. Finally, if the plane intersects one of the pieces of the cone but is NOT parallel to the ``side'' of one of the cones, then an ellipse is generated.