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| Chapter 2 - Conic Sections | |||||
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CHAPTER
2 CONIC SECTIONS LEARNING
OBJECTIVES Upon
completion of this chapter, you should be able to do the following: l.
Determine the equation of a curve using the locus of the equation. 2.
Determine the equation and properties of a circle, a parabola, an ellipse, and
a hyperbola. 3.
Transform polar coordinates to Cartesian coordinates and viceversa. INTRODUCTION This
chapter is a continuation of the study of analytic geometry. The figures
presented in this chapter are plane figures, which are included in the general
class of conic sections or simply "conics." Conic sections are so
named because they are all plane sections of a right circular cone. A circle is
formed when a cone is cut perpendicular to its axis. An ellipse is produced
when the cone is cut obliquely to the axis and the surface. A hyperbola results
when the cone is intersected by a plane parallel to the axis, and a parabola
results when the intersecting plane is parallel to an element of the surface.
These are illustrated in figure 2-1. When
such a curve is plotted on a coordinate system, it may be defined as follows: A
conic section is the locus of all points in a plane whose distance from a fixed
point is a constant ratio to its distance from a fixed line. The fixed point is
the focus, and the fixed line is the directrix.
Figure 2-1.-Conic sections. The
ratio referred to in the definition is called the eccentricity (e). If the eccentricity is greater than 0 and less
than 1, the curve is an ellipse. If e is greater
than 1, the curve is a hyperbola. If e is equal
to 1, the curve is a parabola. A circle is a special case having an
eccentricity equal to 0. It is actually a limiting case of an ellipse in which
the eccentricity approaches 0. Thus, if
The
eccentricity, focus, and directrix are used in the algebraic analysis of conic
sections and their corresponding equations. The concept of the locus of an
equation also enters into analytic geometry; this concept is discussed before
the individual conic sections are presented.
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