Equation Section (Next)Conic Sections A conic section is the structure formed by the intersection of a plane with a double right circular cone, depending on the angle of incidence of the plane with the cone. The structures formed by this intersection are: circles, ellipses, parabolas, ad hyperbolas. The study of conic sections can be traced back to approximately 100 BC, to the Greek mathematician Apollonius of Perga who developed theories on conic sections without the benefit of algebra or analytic geometry. In general a conic section can be expressed by an equation of the form: Ax By Cx Dy F 0, Where the relation defines more than a single point. If A B, then the relation describes a circle. If A and B have the same sign buta B, then the relation describes an ellipse. If exactly one of A or B is equal to zero, then the relation describes a parabola. If A and B have different signs, then the relation describes a hyperbola. Circles and ellipses A circle is the set of all points x, y a fixed distance from a center point, h k. The fixed distance from the central point is the radius, r, of the circle. This distance is given by the equation: xh yk r (1.1) Thus this equation defines the circle. The usual rendering of equation 0.1 as a circle squares both sides of the equation to yield: xh yk r. (1.) Equation 1. can be rewritten in a more general standard form: xh yk r. (1.3) r 1 Example 1.1: The equation x y,3. Realize that x y 3 16 3 16 describes a circle of radius 4 centered at written in general conic form appears as : x y 4x6y3 0. (1.3) The equation x y 3 16 x y3 rewritten in this format becomes: 16 16 1
Equation 1.4 suggests that a twist could be introduced to this scheme. What if the denominators were not equal? Then you would have something of the form xh yk 1. (1.4) This last equation describes an ellipse centered at hk, with an axis of length a in the x- x y direction and an axis of length b in the y-direction. For example 1 is an ellipse centered at 0,0with axes of length 3 and in the x- and y-directions, respectively. It would look like: y 4 0-4 - 0 4 x - -4 NOTE: It would hideously irresponsible of me to suggest improper vocabulary, but I do tend to be hideously irresponsible. Equation 1.4 for the circle highlights that the radius is the same in the x-direction and the y-direction as it should be, since it is a radius. Equation 1.5 could be interpreted to show that the radius in the x-direction is different that the radius in the y- direction. This interpretation might suggest terms like x-radius and y-radius to discuss the attributes of an ellipse, linking back to contrasts/comparisons with a circle --- but that would be hideously irresponsible.
Hyperbolas From above we have xh yk 1 which describes a circle if and an ellipse if. We can introduce a new wrinkle into the equation by changing the addition to a subtraction to get xh yk 1 (.1) Equation.1 gives the general format for the rectangular form of a relation describing a hyperbola. In equation.1, the hyperbola is directed opening bilaterally in the x-direction, h, k. A central rectangle of length a in the x-direction and b in the y-direction centered at may be thought to exist in the middle of the hyperbola, also centered at h, k. The diagonals of this central rectangle, running through it s the corners located at ha, k b and ha, k b, give us the asymptotes of the hyperbola. Had the terms in equation.1 been interchanged, the hyperbola would have opened bilaterally in the y-direction. x y For example, consider 1. The corners of the central rectangle are 3,, 3,, 3,, 3,. The equations of the two diagonals are y x (in blue) and 3 y x (in red). Since the x-variable term is positive, the hyperbola plotted in black is 3 oriented along the x-axis.
Let us consider a more difficult example: 4x 3y 16x 1y 16 0 Note that the coefficient of the x term is negative, while the coefficient of the y term is positive. This tells us 1) the structure is a hyperbola, and ) the hyperbola will be oriented along the y-axis. In order to tackle this equation we first organize the x-terms and the y-terms: 4x 9y 16x36y1604x 16x9y 36y16 0 We next insert execute the complete-the-square algorithm for the x-terms and the y-terms. First complete the square and factor for the y-terms: 4x 16x 9y 36y 16 0 4x 16x 9y 36y3616 360 4x 16x 9y 36y36 16 360 4x 16x 9 y 4y4 16 360 4x 16x 9 y 16 36 0 Now do the same for the x-terms, keeping in mind that the x-terms will eventually have a negative 1 factored out: x x y x x y x y x y 4x 16x 9 y 16 36 0 4x 16x 16 9 y 16 16 36 0 4 16 16 9 1616360 4 4 4 9 1616360 4 9 1616360 4 9 360 Rearrange this last equation by subtracting the constant term from both sides of the equation, and dividing both sides of the equation by the least common multiple of the coefficients: x y x y x y 4 9 360 4 9 36 1 This last equation indicates a hyperbola oriented along the y-axis, centered at,.