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THE ELLIPSE An ellipse is a conic section with an
eccentricity greater than 0 and less than 1. Referring to figure 211, let PO =a FO =c OM = d where F is the focus, O is the center, and P and P' are
points on the ellipse. Then from the definition of eccentricity,
Figure 211.Development of focus and directrix.
and,
Subtraction and addition of the two equations give
Place the center of the ellipse at the origin so that one
focus lies at (  ae,0) and one directrix is the line x =  ale. Figure 212 shows a point on the Y axis that
satisfies the conditions for an ellipse. If P"O=b and FO =c then By definition, e is the
ratio of the distance of P" from the focus and the directrix, so
Figure 212.Focus, directrix, and point P".
or
Multiplying both sides by ale gives
or
so _{
} Now combining equations (2.4) and (2.5) gives
or
Refer to figure 213. If the point
(x,y) is on the ellipse, then the ratio of its distance from F to its distance
from the directrix is e. The distance from (x,y) to the focus (  ae,0)
is
and the distance from (x,y) to the directrix x =
is
The ratio of these two distances is equal to e, so (x +
ae)^{2} + y2
Figure
213.The ellipse.
or
Squaring and expanding both sides gives
Canceling like terms and transposing terms in x to the
lefthand side of the equation gives _{
} Removing a common factor gives
Dividing both sides of equation (2.7) by the righthand
member gives
From equation (2.6) we obtain
so that the equation becomes
This is the equation of an ellipse in standard form. In
figure 214, views A and B, a is the length of the semimajor
axis and b is the length of the semimanor axis. The curve is symmetrical with respect to the X and Y axes,
so you can easily see that figure 214, view A,
has another focus at (ae,0) and a corresponding directrix, x
= a/e. The curve also has vertices at ( a,0). The distance from the center through the focus to the
curve is always designated a and is called the semimajor axis. This
axis may be in either the x or y direction. When it is in the y direction, the
directrix is a line denoted by the equation
Figure 214.Ellipse showing axes. In
the case we have studied, the directrix was denoted by the formula
where
k is a constant equal to a/e. The
perpendicular distance from the midpoint of the major axis to the curve is
called the semiminor axis and is
always signified by b. The
distance from the center of the ellipse to the focus is called c. In any
ellipse the following relations are true for a, b, and c:
Whenever
the directrix is a line denoted by the equation y = k, the major axis is in the
y direction and the equation of the ellipse is as follows: _{
} Refer
to figure 214, view B. This curve has
foci at (0, c) and vertices at (0, a). In
an ellipse the position of the a^{2} and b^{2} terms
indicates the orientation of the ellipse axis. As shown in figure 214, views A and B, value a is the
semimajor or longer axis. In
the previous paragraphs formulas were given showing the relationship between a,
b, and c. In the first portion of this
discussion, a formula showing the relationship between a, c, and the
eccentricity was given. These relationships are used to find the equation of an
ellipse in the following example: EXAMPLE: Find the equation of the ellipse with its center at the origin and
having foci at
and an eccentricity equal
to
, SOLUTION: With the focal points on the X axis, the ellipse is oriented as in
figure 214, view A, and the standard
form of the equation is
With the center at the origin, the numerators of the
fractions on the left are x^{2} and y^{2}. The problem is to
find the values of a and b. The distance from the center to either of the foci is
equal to c (fig. 214, view A), so in this problem
from the given coordinates of the foci. The values of a, c, and e (eccentricity) are related by c=ae or ^{} From the known information, substitute the values of c and
e,
and
Then, using the formula
or _{
} and substituting for a^{2} and c^{2},
gives the final required value of b^{2} = 25 Then the equation of the ellipse is
