MATH10000 Mathematical Workshop Project 2 Part 1 Conic Sections The aim of this project is to introduce you to an area of geometry known as the theory of conic sections, which is one of the most famous and applicable in all of mathematics. The subject has an extremely long and interesting history, beginning with the Greeks over 2,000 years ago. We shall work in the more modern context of coordinate geometry, which was first used by Descartes in the 17th century, but we will refer back regularly to the original viewpoint. The Greeks began with a double cone in 3 dimensional space, making an angle β with the vertical axis for some 0 < β < π/2. They then considered the effect of intersecting the cone with a plane, which is assumed to make an angle of α with the same axis. The case α = π/2 gives rise to a circle. As we proceed, the following cases will emerge: β < α < π/2 an ellipse α = β a parabola (0.1) 0 α < β a hyperbola. 1
From a more algebraic viewpoint, conic sections provide a geometrical method of classifying quadratic equations. In other words, they determine a procedure for identifying any such equation as one of a small number of characteristic types. The most general quadratic equation in two variables x and y takes the form Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, (0.2) where A,..., F are constants. As we proceed, we shall encounter the following facts, which are expressed in terms of the discriminant = B 2 4AC. The equation (0.2) represents an ellipse when < 0 a parabola when = 0, (0.3) a hyperbola when > 0 although certain degenerate cases must also be taken into account. These include the empty set, a single point, a straight line, and a pair of straight lines. You should try to visualise how these might arise from (0.1) above. Once you are familiar with the basic details, you can begin to study practical applications of conic sections. One of the most famous is to the motion of particles (such as planets!) within a gravitational field obeying an inverse square law of attraction - or repulsion. The orbits in such a system are conic sections, as Isaac Newton famously showed in 1713. Without this analysis, satellites and spacecraft could not be launched successfully today. Sources There are, quite literally, thousands of informative references to conic sections in print and on the web. Googling conic sections will reveal enough of these to keep everybody happy (1,100,000, as I write!), although we should all be careful to ensure that any mathematics we take from the web is of reputable origin - there is a lot of misleading stuff out there. To experiment with conic sections you can use the Java Applet at http://cs.jsu.edu/mcis/faculty/leathrum/mathlets/conics.html Bounded Conic Sections The Circle We begin with a familiar geometric definition of the circle, and discuss its description by means of an equation in the (x, y)-plane R 2. 2
Definition A circle with centre C and radius r is the set of points in the plane which are the given distance r from the fixed point C. For any point P = P (x, y) on the circle, the distance P C is always r. So Pythagoras Theorem tells us that we may write the circle more precisely as {(x, y) R 2 : (x h) 2 + (y k) 2 = r 2 }, where the centre is the point with coordinates (h, k). Every circle is bounded, because all of its points lie within a finite distance of the origin. The Ellipse An ellipse is a form of generalised circle, and is correspondingly more difficult to analyse. In fact the theory of the ellipse gives rise to some of the most sophisticated mathematics yet discovered! Definition An ellipse with foci F 1 and F 2 is the set of points in the plane, the sum of whose distances from F 1 and F 2 is constant. So an ellipse whose foci coincide is just a circle. It is usual to begin studying the ellipse in the plane R 2 by assuming that F 1 and F 2 are the points ( c, 0) and (c, 0) respectively, where c 0. 3
Any point P = P (x, y) on such an ellipse obeys P F 1 + P F 2 = 2a, for some positive real number a. Pythagoras Theorem then tells us that the co-ordinates of P satisfy (x + c)2 + y 2 + (x c) 2 + y 2 = 2a, (0.4) and after a considerable amount of algebraic manipulation, this reduces to x 2 a + y2 = 1. (0.5) 2 a 2 c2 Since P F 1 + P F 2 must exceed F 1 F 2, we know that 2a > 2c, and therefore that a 2 c 2 > 0. So we may substitute b = a 2 c 2 into (0.5), and obtain x 2 a + y2 = 1. (0.6) 2 b2 Notice that b < a always holds. The major axis of the ellipse is the segment of the x-axis defined by a x a, and the minor axis is the segment of the y-axis defined by b y b. Because we always have a > c 0, the quantity e = c/a satisfies 1 > e 0; it is called the eccentricity of the ellipse. It also follows that a/e > a, so the entire ellipse lies between the two straight lines x = ±a/e; they are called the directrices. Given any point P (x, y) on the ellipse, we let N (which varies with P, of course!) denote the nearest point on one of the directrices. 4
Proposition For any P, the relationship P F = e P N holds, where F denotes the focus nearer to N. Proof so Let F 1 = (c, 0) and F 2 = ( c, 0). By (0.5), the coordinates of P satisfy y 2 = (a 2 c 2 ) a2 x 2 a 2, P F 1 = (x c) 2 + y 2 = a c a x and P F 2 = (x + c) 2 + y 2 = a + c a x. that is ( a ) P F 1 = a ex = e e x ( a ) = e P N 1 and P F 2 = a + ex = e e + x = e P N 2 respectively, where N i is the nearest point to P on the directrix nearer F i, for i = 1, 2. We may think of this result as giving an alternative definition for the ellipse. The centre of an ellipse is the intersection of its axes of symmetry; so far, this has always been at the origin. As with the circle, we must also consider the possibility of it being at an arbitrary point (h, k). To study this case, we begin by assuming that the foci are the points (h + c, k) and (h c, k). It then follows (as above), that the corresponding ellipse is given by { (x, y) R 2 : } (x h)2 (y k)2 + = 1. a 2 b 2 We may reinterpret this formula by making the substitutions in which case it becomes the more familiar x = x + h and y = y + k, (0.7) (x ) 2 a 2 + (y ) 2 b 2 = 1. (0.8) This is because (0.7) describes what happens to the coordinates of a point in R 2 when we replace the original coordinate axes by new x - and y -axes, whose origin lies at (h, k). In other words, we translate the coordinate system. In so doing, we simplify the equation of the conic, and its type and details may be read off immediately from the result, as in (0.8). 5