Ellipse
Definition and construction

Constructions of an ellipse
Equation of the ellipse, standard equation of the ellipse

Definition and construction, eccentricity and linear eccentricity
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
According to definition we will explain two constructions of the ellipse:
 -  Fasten the ends of a string of length 2a > 2c at two distinct points F1 and F2. Pull the loop of string tight using the pencil until a triangle is formed with the pencil and the two foci as vertices. Keeping the string  pulled tight, move the pencil around until the ellipse is traced out.
-  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 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.
It follows from the equation that an ellipse is defined by values of a and b, or as they are associated through
the relation
a2 - c2 = b2,  we can say that it is defined by any pair of these three quantities.
 Intersections of an ellipse and the coordinate axes we determine from equation by putting,
y = 0  =>   x = + a, so obtained are vertices at the ends of the major axis A1(-a, 0) and A2(a, 0), and
x = 0  =>   y = + b, obtained are co-vertices, the endpoints of the minor axis B1(0, b) and B2(0, -b).
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 which equals the absolute value of ordinates of the points of the ellipse whose abscissas
x = c or x = -c that is
 so the length of the latus rectum the length of the semi-latus rectum of the ellipse.
Pre-calculus contents H