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| Conic
Sections |
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Parabola
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Transformation of the equation of a parabola
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Equation of a translated parabola - the standard form
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Parametric equations of the parabola
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Equations of the parabola examples |
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| Transformation of the equation of a parabola
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| 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.
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| For
p >
0
the parabola opens up, if
p <
0
the parabola opens down as shows
the below figure. |
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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.
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| Equation of a translated parabola - the standard form
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| By parallel shifting of the parabola
y2 = 2px in the direction of the coordinate axes the vertex of the parabola
can be brought at a point A(x0, y0) while coordinates
x
and y of every point of the parabola changes by the |
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value of the
translations x0 and
y0, so that equation of the translated
parabola is
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If p >
0
the parabola
opens to the right and if p
< 0
the
parabola opens to the left.
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If the focal parameter
p = 2a
then the above equation
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becomes,
(y
-
y0)2 =
4a(x
-
x0).
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The translated parabola with the axis parallel to the
x-axis
can also be written as
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x =
ay2
+ by
+ c.
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| Equations of the parabola written in the general form |
| a) the axis of the parabola parallel to the
x-axis
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b) the axis of the parabola parallel to the
y-axis |
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Ay2
+ Bx + Cy +
D =
0, A and
B not
0, |
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Ax2
+ Bx + Cy +
D =
0, A and
C not 0 |
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or x =
ay2
+ by
+ c,
a not 0. |
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or
y =
ax2
+ bx
+ c,
a not 0. |
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| Note
that the parabola has equation that contains only one squared term. |
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| Parametric equations of the parabola
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| Parametric equations of the parabola
y2 =
4ax
with the vertex A
at the origin and the focus F(a,
0), and of its translation (y
-
y0)2 =
4a(x
-
x0)
with the vertex A(x0, y0)
and the focus F(x0
+ a, y0)
written |
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are, |
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| Parametric equations of the parabola
x2 =
4ay
with the vertex A
at the origin and the focus F(0,
a), and of its translation
(x
-
x0)2 =
4a(y
-
y0)
with the vertex A(x0, y0)
and the focus F(y0,
y0
+ a) written |
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are, |
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| Equations
of the parabola examples |
| Example:
Write equation of the parabola
y2 = 2px
passing through the point P(-4,
4) and find the focus,
the equation of the directrix and draw its graph. |
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Solution:
The coordinates of the point P
must satisfy the equation of the parabola
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P(-4,
4)
=>
y2 =
2px
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42 = 2p(-4)
=> p =
-2
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thus, the
equation of the parabola y2 =
-4x.
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The coordinate
of the focus,
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since F(p/2,
0) then
F(-1,
0).
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The equation of the directrix,
as x =
- p/2,
x =
1.
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| Example:
Into a parabola y2 =
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
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As the point P
lies on the parabola then
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The area of the equilateral triangle we express by
coordinates of
P
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| Show
the parameter of the parabola by the side of the triangle, |
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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 y2 =
2px or
y2 = 9x. |
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| Conic
sections contents |
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