Conic Sections
Parabola
Transformation of the equation of a parabola

Parametric equations of the parabola
Equations of the parabola examples
Transformation of the equation of a parabola
The equation  y2 = 2pxp < 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.
Equation of a translated parabola - the standard form
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
value of the translations x0 and y0, so that equation of the translated parabola is
 (y - y0)2 = 2p(x - x0)
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 - y0)2 = 4a(x - x0).
The translated parabola with the axis parallel to the x-axis  can also be written as
x = ay2 + by + c.
Equations of the parabola written in the general form
 a) the axis of the parabola parallel to the x-axis b) the axis of the parabola parallel to the y-axis Ay2 + Bx + Cy + D = 0,  A and B not 0, Ax2 + Bx + Cy + D = 0,  A and C not 0 or   x = ay2 + by + c,  a not 0. or   y = ax2 + 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 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
 respectively are,
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
 respectively are,
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.
 Solution:   The coordinates of the point P must satisfy the equation of the parabola P(-4, 4)   =>    y2 = 2px 42 = 2p(-4)   =>   p = -2 thus, the equation of the parabola   y2 = -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 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 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, -2Ö3pso that,  P(27, 9Ö3) and P'(27, -9Ö3).
Therefore, the equation of the parabola  y2 = 2px  or  y2 = 9x.
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