Conic Sections
    Ellipse
      Definition and construction
         Eccentricity and linear eccentricity
         Constructions of an ellipse
      Equation of the ellipse, standard equation of the ellipse
         Major axis, minor axis, and vertices
         The focal parameter, latus rectum
         Ellipse examples
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.
Ellipse examples
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:  From given quantities of an ellipse determine remaining unknown quantities and write equation of the ellipse,
Solution:    a)  Using
therefore, the semi-minor axis
the linear eccentricity
the semi latus rectum and the equation of the ellipse

the eccentricity and the equation of the ellipse

the semi latus rectum and the equation of the ellipse

      d) unknown quantities expressed through given values,
Example:   Find the equation of the ellipse whose focus is F2(6, 0) and which passes through the point A(53, 4).
Solution:   Coordinates of the point A(53, 4) must satisfy equation of the ellipse, therefore
thus, the equation of the ellipse
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.
Conic sections contents
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