| Three conic sections will be discussed here: the parabola, the ellipse, and the hyperbola. Each of these figures are called conics because they can be formed by intersecting a plane with a cone....hence, the name, conic section. We will look at each conic separately.
The ParabolaIn intermediate algebra (and in the first part of this course) we looked at parabolas with emphasis on the vertex, the intercepts (both x & y), the domain and the range. In order to make the vertex stand out in the equation we completed the square on the quadratic term to yield one of the following formulas: y = a (x - h) 2 + k (for a vertical parabola) x = a (y - k) 2 + h (for a horizontal parabola) where (h,k) was the vertex. If a > 0 then the parabola opened in a positive direction (up or to the right). If a < 0 then the parabola opened in the negative direction ( down or to the left).
The treatment of parabolas discussed below, however, highlights two different aspects than is highlighted in the intermediate algebra approach described above. So, we write the parabolic equation in the form that will highlight these new points. Before we get to that new form we need a little more information about parabolas.
What makes a curve a parabola and not just a bent line? I don't imagine that you question the fact that every enclosed curve is not a circle. That is, you would not call a teardrop shape a circle. But what is the geometric definition of a circle that prevents the figure from being classified as a circle? Geometric definition of a circle: The set of all points in a plane that are equidistant from a set point called the center.
Geometric definition of a parabola: the set of all points in a plane that are equidistant from a set point (called the focus) and a set line (called the directrix). The standard form of a parabola that we are now going to use helps us to find the focus and the directrix. Note: The focus and directrix will not help you get a better sketch of the parabola than you have gotten in the past. The focus and directrix simply give us more information about the soul of a parabola.
Notice that the left side of the equation has the squared term isolated.
Your text has number of other formulas that tell you what to do with the p in order to find the focus and the directrix. While the formulas do work, I find it easier to draw a picture of the parabola and use it to guide me in the use of p to determine the focus & directrix as Example 1 demonstrates.
Example 1: Given the standard form for the parabola: (x - 3)2 = - 8 (y + 2), find the vertex, value of p, focus, and directrix. Solution: Here is a sketch of what we have so far
Example 2: Find the standard form of the parabola that has a focus of ( - 2 , 8) and directrix x = 4
If you haven't already read what your text has to offer on the wonderful world of parabolas then do so now. Your life just won't be complete until you get the chance to hear from another expert source on this fascinating topic.
The EllipseJust like every curve is not a parabola, every squashed circle is not an ellipse. So, what makes thes ellipse so special? The geometric definition again answers our question. The geometric definition of an ellipse: The set of all points in a plane whose sums of distances between the point and the two fixed points (the foci) is a constant. The longer axis that passes through the foci and the center with endpoints is called the major axis and the shorter axis that goes through the center and is perpendicular to the major axis is called the minor axis. Your text has some nice diagrams of ellipses with the the key points highlighted.
The standard form a an ellipse highlights the characteristics of the ellipse that make it unique. That is, the standard form will give us the information needed to find the center, the foci, the endpoints of the major axis, and the endpoints of the major axis.
a = distance from the center to the endpoint of the major axis
Orientation of the parabola:You tell whether the ellipse is horizontal or vertical by the location of the largest denominator. If the number under x is largest then you have a horizontal ellipse (major axis is parallel to the x-axis). If the number under y is the largest then you have a vertical ellipse (major axis is parallel to the y-axis. In short, a > b.
To find the focus:First, we need to find the distance from the center to one of the foci. We will let c represent that distance. The formula that relates a, b, and c is the following: a2 = b2 + c2.
Your textbook has a much fuller explanation of how we get the equation defining the relationship between a,b, and c. Here I am just trying to summarize some of the main points from the section, not take its place. So, be sure that read through your book as well as this summary.
Example 3: Given the standard form of an ellipse, state the center, foci, endpoints of the major axis, and endpoint of the minor axis.
Solution:
Example 4: Find the equation for an ellipse with foci (-3,2) & ( 5, 2), length of the minor axis of 10.
The ellipse (as well as the parabola) has terrific reflective properties that make it interesting to doctors, engineers, and math instructors. For more enlightenment on this many uses of an ellipse, please visit the MCC Information Commons and check out the video # 12 in the popular series College Algebra: In Simplest Terms.
The HyperbolaFinally, we come to the hyperbola. This conic is probably the one with which you have the least amount of experience. The graph of a hperbola looks like two parabolas that have been placed nose to nose, but, of course, we know that there must be some geometric charactertistics of the hyperbola that makes it different than just a double parabola.
The geometric definition of a hyperbola: The set of all points in a plane for which the difference between distances from the point to two fixed points (the foci) is a positive constant. The line segment that goes through the center and the foci with the vertices as endpoints is called the transverse axis. The conjugate axis runs through the center and is perpendicular to the transverse axis.
General information that will interest us about the hyperbola will include the center, the vertices, and the foci. To help us in graphing the hyperbola we will also want to know the endpoints of the conjugate axis. And, we will be interested in two lines, called the asymptotes of the hyperbola, which help us to describe the curve of the hyperbola by using linear equations. The standard form of the hyperbola will help us to determine the desired information about the hyperbola.
a = distance from the center to the endpoint of the transverse axis
Orientation of the hyperbola: You tell whether the transverse axis of the hyperbola is horizontal or vertical by the sign ( + or - ) of the terms. If the coefficient of the x-term is positive, as in the first equation above, then the transverse axis is parallel to the x-axis (horizontal). If the coefficient of the y-term is positive, as in the second equation above, then the transverse axis is parallel to the y-axis (vertical). a2 is always associated with the term that has the positive coefficient. For the hyperbola we do not care if a > b or b > a (unlike the ellipse, where a > b).
To find the focus: First, we need to find the distance from the center to one of the foci. We will let c represent that distance. The formula that relates a, b, and c. is the following: c2 = a2 + b2. Don't forget to READ YOUR TEXT!! Your textbook has a much fuller explanation of how we get the equation defining the relationship between a,b, and c. Here I am just trying to summarize some of the main points from the section, not take its place. So, be sure that read through your book as well as this summary.
Asymptotes: The asymptotes are lines that serve as guides to your curve. As you trace further and further out on the tails of your hyperbola the distance between the curve of the hyperbola and the line of the asymptote will get smaller. Ultimately, you will cease to be able to tell the difference between the two. That is, at the ends of the hyperbola, the lines of the asymptotes will describe the same points as the points of the curve. So, we say that the asymptote describes the curve of the hyperbola.
Example 5: Find the asymptotes for the hyperbola:
y = 1.41 x - 10.24 and y = -1.41 x -1.76
WHY does this work? An intuitive explanation for why the above procedure works for finding asymptotes is based on the general idea of what these asymptotes represent. These lines describe the relationship between the x-values and the y-values on the hyperbolic curve when the magnitudes of x & y get very large. That is, they describe the relationship between x & y at the end of the graph. When the magnitudes of both x and y get very large the term '1' will have an insignificant effect on the relationship between x & y. So we replace '1' with '0' and then simplify the remaining relationship that is described by the equation.
Other problems that involve hyperbolas: The other types of problems that involve hyperbolas are pretty similar to the examples that I have already given with parabolas and ellipses, so I won't go into additional specific examples. Read through the examples already given and then try the ones in your text.
Where are hyperbolas used? Your text includes some examples of uses for hyperbolas and more are included in video #12 of the series College Algebra: In Simplest Terms. One word of caution about the video before you run right over to reserve your copy - the video presents a slightly different approach to defining a, b, and c....so FAST FORWARD OVER THAT PART. Watch the video for the pure enjoyment of learning of the many places hyperbola are used in your world.
Last comment: I realize that I haven't gone over every detail on conics that is addressed by your text - that is intentional. In some cases I am expecting you to extrapolate from what your text and I have presented. In other cases I am hoping to elicit a question or two from you.
© 1997 Jo Steig
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The coefficient of the linear factor on the right side (4p) gives us the additional information needed to find the focus and the directrix.| p | represents the distance between the vertex & the focus and the distance between the vertex & the directrix.
the vertex is (3, -2)
