College Trigonometry, 2011a

11.6 Hooked on Conics Again 981
1. We have A = 21, B =10 √ 3andC =31soB 2 − 4AC =(10 √ 3) 2 − 4(21)(31) = −2304 < 0.
Theorem 11.11 predicts the graph is an ellipse, which checks with our work from Example
11.6.1 number 2.
2. Here, A =5,B =26andC =5,soB 2 − 4AC =26 2 − 4(5)(5) = 576 > 0. Theorem 11.11
classifies the graph as a hyperbola, which matches our answer to Example 11.6.2 number 1.
3. Finally, we have A = 16, B =24andC = 9 which gives 24 2 − 4(16)(9) = 0. Theorem 11.11
tells us that the graph is a parabola, matching our result from Example 11.6.2 number 2.
11.6.2 The Polar Form of Conics
In this subsection, we start from scratch to reintroduce the conic sections from a more unified
perspective. We have our ‘new’ definition below.
Definition 11.1. Given a fixed line L, apointF not on L, and a positive number e, aconic
section is the set of all points P such that
the distance from P to F
the distance from P to L = e
The line L is called the directrix of the conic section, the point F is called a focus of the conic
section, and the constant e is called the eccentricity of the conic section.
We have seen the notions of focus and directrix before in the definition of a parabola, Definition 7.3.
There, a parabola is defined as the set of points equidistant from the focus and directrix, giving an
eccentricity e = 1 according to Definition 11.1. We have also seen the concept of eccentricity before.
It was introduced for ellipses in Definition 7.5 in Section 7.4, and later extended to hyperbolas in
Exercise 31 in Section 7.5. There,e was also defined as a ratio of distances, though in these cases
the distances involved were measurements from the center to a focus and from the center to a
vertex. One way to reconcile the ‘old’ ideas of focus, directrix and eccentricity with the ‘new’ ones
presented in Definition 11.1 is to derive equations for the conic sections using Definition 11.1 and
compare these parameters with what we know from Chapter 7. We begin by assuming the conic
section has eccentricity e, afocusF at the origin and that the directrix is the vertical line x = −d
as in the figure below.
y
d
r cos(θ)
P (r, θ)
r
θ
x = −d
O = F
x