
MATHEMATICS  THEORY



The parabolas, ellipses and hyperbolas will
be introduced in this section followed by the discussion of the conic
sections in polar coordinates.




Parabola
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Parabola: (x  p)^{2} = 4m(y  q)
Parabola: (y  p)^{2} = 4m(x  q)


Parabolas


A parabola is defined as the set of points
in a plane that are equidistant from a fixed point (focus) and a fixed
line (directrix). The line through the focus and perpendicular to the
directrix is referred to as the axis of the parabola. The point on the
axis that is equidistant from the directrix and the focus is referred
to as the vertex of the parabola. The vertex is also the intersection
point of the parabola and the axis.
The standard form of the equation for a parabola with vertex (p, q),
axis x = p, directrix y = q  m and focus (p, q + m) is given by
(x  p)^{2} = 4m(y  q)
The parabola opens upward if p is positive while it opens downward when
m is negative.
Similarly, by interchanging the parameter x and y the parabola will
be rotated 90 degrees, giving
(y  p)^{2} = 4m(x  q)
This parabola has a vertex (p, q), axis y = p, directrix x = q  m and
focus (p, q + m), as shown in the figure. The parabola opens to the right
if m is positive and to the left when p is negative.





Ellipses

Ellipse
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Ellipse with Horizontal Major Axis
Ellipse with Vertical Major Axis


An ellipse is defined as the set of points in a plane such that the sum
of the distances between a point on the ellipse and two fixed points (foci)
is constant. The center of the ellipse is the midpoint of the line segment
between the two foci. The points of intersection of the ellipse with the
line through the foci are referred to as the vertices. The line segment
between the vertices is the major axis. The minor axis is the line segment
through the center and perpendicular to the major axis.
If the foci of an ellipse are on the xaxis, the standard form of the
equation for the ellipse is given by
with center (p, q), vertices (p  a, q) and (p + a, q), foci (p  c,
q) and (p + c, q). Note that a > c > 0, a > b > 0 and c^{2} =
a^{2}  b^{2}.
When the foci of an ellipse are on the yaxis, the equation is given
by:
with center (p, q), vertices (p, q  a) and (p, q + a), foci (p, q 
c) and (p, q + c). Note that a > c > 0, a > b > 0 and c^{2} =
a^{2}  b^{2}.
It is noted from the above two equations that when the foci coincide,
c = 0 and a = b. The ellipse becomes a circle with radius r = a = b.
The eccentricity of an ellipse is defined as
where 0 < e < 1. The ellipse is long and thin when e approaches
1 and the ellipse is a circle when e is close to 0.






Hyperbolas

Hyperbola
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Hyperbola with Horizontal Transverse Axis
Asymptotes for the Hyperbola
Hyperbola with Vertical Transverse Axis


A hyperbola is defined as the set of points in a plane such that the distance
difference between a point on the hyperbola and two fixed points (foci)
is constant. The center of the hyperbola is the midpoint of the line segment
between the two foci. The line segment through the foci is the transverse
axis. The conjugate axis is the line segment through the center and perpendicular
to the transverse axis. The points of intersection of the hyperbola with
the transverse axis are referred to as the vertices. The hyperbola has
two separate parts called the branches.
The standard form of equation for a hyperbola with horizontal transverse
axis is given by
with center (p, q), vertices (p  a, q) and (p + a, q), foci (p  c,
q) and (p + c, q). Note that c > a > 0, c > b > 0 and c^{2} =
a^{2} + b^{2}. A hyperbola will approach, but never reach,
two lines called asymptotes. These lines are
and
For a hyperbola with a vertical tranverse axis, the equation is given
by:
with center (p, q), vertices (p, q  a) and (p, q + a), foci (p, q 
c) and (p, q + c). c > a > 0, c > b > 0 and c^{2} =
a^{2} + b^{2}. The asymptotes for the hyperbola is
and
The eccentricity of a hyperbola is defined the same as that of an ellipse.






Conic Sections in Polar Coordinates



The three types of conic sections (i.e., parabola, ellipse
and hyperbola) can be treated in a more unified form using a focus and
directrix in polar coordinates. The analysis presented here assumes the
focus is at the origin.
Let F be a fixed point called the focus and L be a fixed line called
the directrix in a plane. Let e be the eccentricity and all points P
of a conic section satisfy the equation:
Basically, the eccentricity is given by the ratio of the distance from
F to the distance from L. The conic section is
(1) an ellipse when e < 1,
(2) a parabola when e = 1, and
(3) a hyperbola when e > 1.
The conic section in polar coordinates has the following form:
or
where e is the eccentricity and d is the distance to the directrix. 



