Graphics Reference
In-Depth Information
y
y
Directrix
Directrix
Directrix
(0,b)
F(ae,0)
F¢(-ae,0)
x
x
O
(-a,0)
(a,0)
F(a,0)
O
Ellipse
(0,-b)
Parabola
x + a = 0
x + a/e = 0
x - a/e = 0
(a)
(b)
y
directrix
directrix
y = -bx/a
y = bx/a
O
F¢(-ae,0)
F(ae,0)
x
(-a,0)
(a,0)
Hyperbola
x + a/e = 0
x - a/e = 0
(c)
Figure 3.21.
The natural coordinate systems for conic sections.
section. If
C
is a parabola, then
L
1
is called its
axis
. If
C
is an ellipse, then
L
1
and
L
2
will be called the
major
and
minor (coordinate) axis
of
C
, respectively. If
C
is a
hyperbola, then
L
1
and
L
2
will be called the
transverse
and
conjugate (coordinate) axis
of
C
, respectively. In the case of either an ellipse or hyperbola the point
O
is called its
center
.
Figure 3.21 shows the coordinate systems and coordinate axes for parabolas,
ellipses, and hyperbolas. Ellipses and hyperbolas are called central conic sections
because they have a center. One can show that the conic sections are symmetric about
their axes, meaning that if a point belongs to them, then the reflected point about
their axes will belong to them also.
In the case of the plane
R
2
, we can use Theorem 3.6.1 and write out the constraints
on the distances of points from the focus and directrix in terms of an equation. It is
easily seen to be a quadratic equation. Let us look at the equations for some special
well-known cases in their natural coordinate system.
y
2
The parabola:
= 4ax
(3.32)
Focus:
(a, 0), a > 0
Directrix:
x =-a
Eccentricity:
e = 1
