Vectors, Matrices and Geometry

CHAPTER THREE: CONIC SECTIONS

CHAPTER THREE CONIC SECTIONS The curves known as conic sections comprise the ellipse, hyperbola and parabola. They are, after the circle, the simplest curves. This being so, it is not surprising that they have been known and studied for a long time. Their discovery is attributed to Menaechmus , a Greek geometer and astronomer of the 4th century BC. Like Hipprocrates of the 5th century BC before him, Menaechmus, in attacking the Delian problem of duplication of a cube, found himself facing the task of constructing two mean proportionals x and y between two given line segments of length a and b: a:x=x:y=y:b. From this continued proportion it follows that Hence leading to ., x- = ay and xy = ab . x3 =axy =a 2 b a3 : x3 =a3 : a2 b =a : b . This means that if b is chosen to be 2a, x would be the side of a cube twice the volume of the cube with side a. Menaechmus then recognized that and xy = ab are represented by curves which are cut out of a right circular cone by plane sections. More than a hundred years later Appolonius of Perga, who is also said to be the founder of Greek mathematical astronomy, wrote a most thorough treatise on conic sections. The majority of the written works by Appolonius are now lost. Fortunately of the original eight books of Pis chef-d'oeuve - The Conics, the first four books remains extant in Greek. Three of the next four books also survived in Arabic translation. 89 Vectors, Matrices and Geometry In this chapter we shall use methods of analytic geometry and work with a unified definition of a conic section which is given in terms of the distance to a fixed point and the distance to a fixed straight line in the plane. Later it shall be shown in two different ways that these curves are actually cut out of a right circular cone by plane sections. 3.1 Focus, directrix and eccentricity Instead of treating the three types of conic sections individually as it is usually done in secondary school analytic geometry, we shall take a more coherent view of our subject and begin our investigation with a unified geometric definition of the conic sections. 3.1.1 DEFINITION A conic section or a conic is the locus of a point X which moves in a plane containing a fixed point F and a fixed line D in such a way that the distance IXFI from the point F is in a constant ratio e (e =F 0) to its perpendicular distance IXEI from the straight line D: IXFI = IXEle . The fixed point F is called a focus, the fixed straight line D is called a directrix and the constant ratio e is called the eccentricity of the conic. A conic is called an ellipse, a parabola or a hyperbola accordingly as its eccentricitye is less than, equal to or greater than unity. F Fig 3-1 In the subsequent sections we proceed to derive equation of each type of conics relative to an appropriately chosen pair of coordinate axes. 90 Conic Sections 3.2 The parabola Given a parabola with focus F and directrix D. Then by Definition 3.1.1, the parabola in question consists of all points X in the plane such that IXFI =IXEI where IXEI is the perpendicular distance from X to D. Our immediate task is to lay down a pair of coordinate axes so that the given parabola would be represented by an equation of the simplest form. For this purpose we drop the perpendicular FH from F on D. Then the midpoint 0 of the segment FH is a point on the parabola. We choose 0 as the origin and the ray originating from 0 towards F as the positive x-axis. With the origin and x-axis being chosen, the yaxis is consequently fixed in the plane (see Figure 3-2). IT we denote by 2a > 0 the distance from F to D, then F =(a, 0) and the directrix D is the line x + a =o. Now the distances from a point X =(x, y) in the plane to the focus F and to the directrix D are respectively IXF\ = J(x - a)2 + y2 and IXE\ = Ix + al . Therefore the parabola consists of all points X = (x, y) such that (1) Squaring, we get (2...