
Conic
Sections 


Parabola

Transformation of the equation of a parabola

Equation of a translated parabola  the standard form

Parametric equations of the parabola

Equations of the parabola examples 





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.





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 not
0, 

Ax^{2}
+ Bx + Cy +
D =
0, A and
C not 0 
or x =
ay^{2}
+ by
+ c,
a not 0. 

or
y =
ax^{2}
+ bx
+ c,
a not 0. 

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, 






Equations
of the parabola examples 
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:
Into a parabola y^{2} =
2px
inscribed is an equilateral triangle whose one vertex coincides with the
vertex of the parabola and whose area A =
243Ö3.
Determine equation of the parabola and remaining vertices of the triangle. 
Solution:
Let write coordinates of a point
P
of the
parabola as elements of the equilateral triangle


As the point P
lies on the parabola then


The area of the equilateral triangle we express by
coordinates of
P





Show
the parameter of the parabola by the side of the triangle, 


and
the vertices of the triangle P(6p,
2Ö3p) and
P'(6p,
2Ö3p)
so that, P(27,
9Ö3) and
P'(27,
9Ö3). 
Therefore,
the equation of the parabola y^{2} =
2px or
y^{2} = 9x. 








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