Conic Sections

OVERVIEW In this chapter we give geometric definitions of parabolas, ellipses, and
hyperbolas and derive their standard equations. These curves are called conic sections, or
conics, and model the paths traveled by planets, satellites, and other bodies whose motions
are driven by inverse square forces. In Chapter 13 we will see that once the path of a moving
body is known to be a conic, we immediately have information about the body’s velocity
and the force that drives it. Planetary motion is best described with the help of polar coordinates,
so we also investigate curves, derivatives, and integrals in this new coordinate
system.
685
CONIC SECTIONS
AND POLAR COORDINATES
Chapter
10
Conic Sections and Quadratic Equations
In Chapter 1 we defined a circle as the set of points in a plane whose distance from some
fixed center point is a constant radius value. If the center is (h, k) and the radius is a, the
standard equation for the circle is It is an example of a conic
section, which are the curves formed by cutting a double cone with a plane (Figure 10.1);
hence the name conic section.
We now describe parabolas, ellipses, and hyperbolas as the graphs of quadratic equations
in the coordinate plane.
Parabolas
sx - hd2 + s y - kd2 = a2 .
10.1
DEFINITIONS Parabola, Focus, Directrix
A set that consists of all the points in a plane equidistant from a given fixed point
and a given fixed line in the plane is a parabola. The fixed point is the focus of
the parabola. The fixed line is the directrix.
If the focus F lies on the directrix L, the parabola is the line through F perpendicular
to L. We consider this to be a degenerate case and assume henceforth that F does not lie
on L.
A parabola has its simplest equation when its focus and directrix straddle one of the
coordinate axes. For example, suppose that the focus lies at the point F(0, p) on the positive
y-axis and that the directrix is the line y = -p (Figure 10.2). In the notation of the figure,