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Analytic geometry - Conic sections
 Ellipse
 ::  Ellipse, definition and construction, eccentricity and linear eccentricity

 By intersecting either of the two right circular conical surfaces (nappes) with the plane inclined to the axis of the cone at a greater angle than that made by the generating segment or generator (the slanting edge of the cone), i.e., when the plane cuts all generators of a single cone, the resulting curve is the ellipse e.

An ellipse is the set of points (locus) in a plane whose distances from two fixed points have a constant sum. The fixed points F1 and F2 are called foci. Thus, the sum of distances of any point P, of the ellipse, 

from the foci is    | F1P | + | F2P | = 2a,  and the distance between foci    | F1F2 | = 2c,

where 2a is the major axis, 2c is called focal distance and c is also called linear eccentricity.

Quantities a and c,  a > c,  uniquely determine an ellipse. The ratio e = c/a,  e < 1 is called eccentricity of the ellipse.
The ellipse can also be defined as the locus of points the ratio of whose distances from the focus to a vertical line, known as the directrix ( d ), is a constant e, where
Constructions of an ellipse
On a given line segment A1' A2'  = 2a, of the major axis, choose an arbitrary point R.

Draw an arc of the radius A1'R = r1 with center F1 and then draw an arc with center F2 and radius A2'R = r2, intersecting the arc at points P1 and P2.

Repeat the same procedure by drawing the arc of the radius r1 centered at F2 and the arc of the radius r2 with the center F1 to obtain intersections P3 and P4. 

Using this method we can draw as many points of the ellipse as needed, noticing that while choosing point R, always must be r1 > a - c and r2 > a - c.

This construction shows that the ellipse has two axes of symmetry of different length, the major and minor axes. Their intersection is the center of the ellipse.
Equation of the ellipse, standard equation of the ellipse
If in the direction of axes we introduce a coordinate system so that the center of the ellipse coincides with 
the origin, then coordinates of foci are
F1(-c, 0) and F2( c, 0).
For every point P(x, y) of the ellipse, according to the
definition  r1 + r2 = 2a, it follows that
after squaring
and reducing
Repeated squaring and grouping gives
(a2 - c2) · x2 + a2y2 = a2 · (a2 - c2),
    and since     a2 - c2 = b2
  follows    b2x2 + a2y2 = a2b2     equation of the ellipse,
and after division by  a2b2, standard equation of the ellipse.
The line segments A1A2 = 2a and B1B2 = 2b are the major and minor axes while a and b are the
semi-major and semi-minor axes respectively. So the arc of the radius a centered at B1and B2 intersects the major axis at the foci F1and F2.

The focal parameter, called latus rectum and denoted 2p, is the chord perpendicular to the major axis
passing through any of the foci, as shows the above figure.

The length of the latus rectum the length of the semi-latus rectum of the ellipse.
Example:  Given is equation of the ellipse 9x2 + 25y2 = 225, find the lengths of semi-major and semi-minor axes, coordinates of the foci, the eccentricity and the length of the semi-latus rectum.
Solution:  From the standard equation we can find the semi-axes lengths dividing the given
equation by 225,
coordinates of the foci F1(-c, 0) and F2( c, 0), since
Example:  Write equation of the ellipse passing through points A(-4, 2) and B(8, 1).
Solution:  Given points must satisfy equation of the ellipse, so
Therefore, the equation of the ellipse or   x2 + 16y2 = 80.
The parametric equations of the ellipse
Equation of the ellipse in the explicit form help us to explain another construction

of the ellipse. So, in the coordinate system draw two concentric circles of radii equal to lengths of the semi axes a and b, with the center at the origin as shows the figure.

An arbitrary chosen line through the origin intersects the 
circle of the radius a at the point R and the circle of radius b at M. 

Then, the parallel line with the major axis through M intersects the parallel line with the minor axis through R, at a point P(x, y) of the ellipse. Proof,
in the figure, OS = x, PS = y and

as the triangles OMN and ORS are similar, then

OM : OR  = MN : RS  or  b : a = PS : RS,
so that
It proves that the point P(x, y) obtained by the construction lies on the ellipse. This way, using the figure, we also derive
the parametric equations of the ellipse where the parameter t is an angle 0 < t < 2p.

By dividing the first parametric equation by a and the second by b, then square and add them, obtained is standard equation of the ellipse.
Equation of a translated ellipse

The equation of an ellipse that is translated from its standard position can be obtained by replacing x by

 x0 and y by y0 in its standard equation,

The above equation can be rewritten into  Ax2 + By2 + Cx + Dy + E = 0.

Every equation of that form represents an ellipse if A not equal B and A · B > 0 that is, if the square terms have unequal coefficients, but the same signs.

Example:  Given is equation of the ellipse 4x2 + 9y2 + 24x -18y + 9 = 0,  find its center S(x0, y0), the semi-axes and intersections of the ellipse with the coordinate axes.
Solution:  Coordinates of the center and the semi-axes are shown in the equation of the translated ellipse,
Rewrite the given equation to that form,
4(x2 + 6x) + 9(y2 - 2y) + 9 = 0
     4[(x + 3)2 - 9] + 9[(y -1)2 -1] + 9 = 0
                            4(x + 3)2 + 9(y -1)2 = 36  or
therefore,  S(-3, 1),  a = 3 and b = 2.

Intersections of the ellipse and the x-axis we obtain by setting  y = 0 into the equation of the ellipse, thus

4x2 + 24x + 9 = 0,    x1,2 = -3 ± 3Ö3/2,

and intersections of the ellipse with the y-axis by setting  x = 0,   =>    9y2 -18y + 9 = 0,    y1,2 = 1.
 
 
 
 
 
 
 
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