|
Conic
Sections |
|
|
Parabola
|
Definition and construction
of the parabola
|
Construction of the parabola
|
Vertex form of the equation of a parabola
|
Transformation of the equation of a parabola
|
|
|
|
|
|
Definition and construction
of the parabola
|
A parabola is the set of all points in a plane that are the same distance from a point F, called the focus, and a
line d, called the
directrix. |
A parabola is uniquely determined by the distance of
the focus from the directrix.
|
This distance is called the
focal parameter
p
and its
midpoint A is the vertex
(apex) of the parabola.
|
The parabola has an axis of symmetry which passes
through the
focus perpendicular to the directrix.
|
The distance of any point
P
of the parabola from the
directrix and from the focus is denoted r, so
|
d(Pd
P) =
FP =
r.
|
Construction of the parabola
|
To a given parameter
p
of the parabola, draw corresponding directrix d
and the focus F.
|
To the distances greater then
p/2
draw lines, of arbitrary dense, parallel to the
directrix.
|
Then, intersect each line at two
symmetric points by arc centered at the focus with a radius
which equals the distance of that line from the
directrix.
|
The distance
p/2
from the vertex A
to the directrix or focus is called focal
distance.
|
|
|
|
|
Vertex form of the equation of a parabola
|
If a parabola is placed so that its vertex coincides with the origin of the coordinate system and its axis lies
along the x-axis then for every point of the parabola |
|
According to
definition of the parabola FP =
d(Pd P) = r
|
|
after squaring |
|
or |
|
y2 =
2px |
the
vertex form of the equation of the parabola, |
|
or |
y2 =
4ax |
if the distance between the directrix and |
|
focus is given by
p = 2a.
|
The explicit form of the equation
y = ±
Ö2px
|
shows that to
every positive value of x
correspond two
opposite values of y
which are symmetric relative to the
x-axis.
|
|
|
|
|
The parabola
y2 =
2px,
p >
0
|
is not defined for
x <
0, it opens to
the right.
|
For x
= p/2
the
corresponding ordinate
y = ± p.
|
This parabola is not a function since the vertical line
crosses
the graph more than once.
|
A focal chord is a line segment passing through the focus
with endpoints on the curve.
|
The latus rectum is the focal chord
(P1P2 =
2p)
perpendicular
to the axis of the parabola.
|
Therefore, we can easily sketch the graph of the parabola
using following points,
|
A(0,
0), P1,2(p/2,
±
p) and P3,4
|
|
|
|
|
Transformation of the equation of a parabola
|
The equation
y2 = 2px,
p < 0
represents the parabola opens to the left since must be y2
> 0. Its axis of
symmetry is the x-axis. |
If variables
x and
y change the role obtained is the parabola whose axis of
symmetry is y-axis.
|
For
p >
0
the parabola opens up, if
p <
0
the parabola opens down as shows
the below figure. |
|
This parabola we often write
y = ax2,
where
a = 1/(2p) , with the focus at
F(0,
1/(4a)) and the directrix |
y =
-1/(4a).
This parabola is a function since a vertical line crosses its graph at only one point.
|
|
|
|
|
|
|
|
|
Conic
sections contents |
|
|
|
Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |