

Analytic geometry
 Conic sections


Parabola

Definition and construction
of the parabola

By intersecting
either of the two right circular conical surfaces (nappes) with the plane parallel to any generator of the cones, then the intersection curve is
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(P_{d}_{
}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 xaxis then for every point of the parabola


shows that to
every positive value of x
correspond two opposite values of y
which are symmetric relative to the xaxis.


The parabola
y^{2} =
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
(P_{1}P_{2
}=
2p)
perpendicular
to the axis of the parabola.

Therefore, we can easily sketch the graph of the
parabola using following points, 
A(0,
0), P_{1,2}(p/2,
±
p) and P_{3,4}(2p,
±2p)




Transformation of the equation of a parabola

The equation
y^{2} = 2px,
p < 0
represents the parabola opens to the left since must be y^{2}
> 0. Its axis of
symmetry is the xaxis.

If variables
x and
y change the role obtained is the parabola whose axis of
symmetry is yaxis.

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 = ax^{2},
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.


Equation of a translated parabola  the standard form

By parallel shifting of the parabola
y^{2} = 2px in the direction of the coordinate axes the vertex of the parabola
can be brought at a point A(x_{0}, y_{0}) while coordinates
x
and y of every point of the parabola 
changes by the value of the
translations x_{0} and
y_{0}, so that equation of the translated
parabola is

(y

y_{0})^{2} = 2p(x

x_{0}) 
If p >
0
the parabola
opens to the right and if p
< 0 the parabola opens to the left.

If the focal parameter
p = 2a
then the above equation

becomes, (y

y_{0})^{2} =
4a(x

x_{0}).

The translated parabola with the axis parallel to the
xaxis can also be written as

x =
ay^{2}
+ by
+ c.





The parabola whose axis of symmetry is parallel to the yaxis

By translating the parabola
x^{2} = 2py
its vertex is moved from the origin to the point A(x_{0}, y_{0})
so that its equation transforms to

(x

x_{0})^{2} = 2p(y

y_{0}).

The axis of symmetry of this parabola is parallel to the
yaxis.
This parabola is a function that we usually
write

y = ax^{2}
+ bx + c
or y

y_{0} =
a(x

x_{0})^{2},

where 


This parabola opens up

if
a >
0
and opens down if
a < 0.




Equations of the parabola written in the general form

a) the axis of the parabola parallel to the
xaxis

b) the axis of the parabola parallel to the
yaxis 
Ay^{2}
+ Bx + Cy +
D =
0, A and
B
are
not
zero, 
Ax^{2}
+ Bx + Cy +
D =
0, A and
C
are
not
zero 
or x =
ay^{2}
+ by
+ c,
a
is
not
zero. 
or
y =
ax^{2}
+ bx
+ c,
a
is
not
zero. 

Note
that the parabola has equation that contains only one squared term. 

Parametric equations of the parabola

Parametric equations of the parabola
y^{2} =
4ax
with the vertex A
at the origin and the focus F(a,
0), and of its translation (y

y_{0})^{2} =
4a(x

x_{0})
with the vertex A(x_{0}, y_{0})
and the focus F(x_{0}
+ a, y_{0})
written

respectively
are, 





Parametric equations of the parabola
x^{2} =
4ay
with the vertex A
at the origin and the focus F(0,
a), and of its translation
(x

x_{0})^{2} =
4a(y

y_{0})
with the vertex A(x_{0}, y_{0})
and the focus F(y_{0},
y_{0}
+ a) written

respectively
are, 






Example:
Write equation of the parabola
y^{2} = 2px
passing through the point P(4,
4) and find the focus,
the equation of the directrix and draw its graph.

Solution: The coordinates of the point P
must satisfy the equation of the parabola

P(4,
4)
=>
y^{2} =
2px

4^{2} = 2p(4)
=> p =
2

thus, the
equation of the parabola y^{2} =
4x.

The coordinate
of the focus,

since
F(p/2,
0)
then F(1,
0).

The equation of the directrix,
as x =
 p/2, x =
1.




Example:
Find the vertex, the focus and the equation of the directrix and draw the graph of the parabola
y =
x^{2}
+ 6x 
7.

Solution: Rewrite
the equation of the parabola in the translatable form

(x

x_{0})^{2} = 2p(y

y_{0}) or
y

y_{0} = a(x

x_{0})^{2}

so, y =
x^{2}
+ 6x 
7
=> y =
(x^{2}

6x) 
7

y =
[(x

3)^{2}

9] 
7

y

y_{0} = a(x

x_{0})^{2},
y

2 = (x

3)^{2},
a
= 1.

The vertex of the parabola
A(x_{0}, y_{0}),
or
A(3, 2).

he focus
F(x_{0},
y_{0 }+ 1/(4a)),
or
F(3,
7/4).

The equation of the directrix,

y =
y_{0 }
1/(4a),
y = 2 +
1/4 or
y =
9/4.


















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