Conic Sections
    Ellipse
      Equation of the ellipse, standard equation of the ellipse
         Major axis, minor axis, and vertices
         The focal parameter, latus rectum
      The parametric equations of the ellipse
         Ellipse examples
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.
The parametric equations of the ellipse
Equation of the ellipse in the explicit form can 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.
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.
Example:  In the ellipse 4x2 + 9y2 = 144 inscribed is a rectangle whose vertices lie on the ellipse and whose sides are parallel with ellipse axes. Longer side, which is parallel to the major axis, relates to the shorter side as 3 : 2. Find the area of the rectangle.
Solution:  It follows from the given condition that the coordinates of vertices of the rectangle must satisfy the 
same ratio, i.e.,      x : y = 3 : 2   =>   x = 3y/2.
To determine points of the ellipse of which coordinates are in this ratio, put these variable coordinates into equation of the ellipse, 
P(3y/2, y)  =>   4x2 + 9y2 = 144
                4(3y/2)2 + 9y2 = 144  =>    18y2 = 144,
y1,2 = 144/18 = 22,    x = 3y/2  =>  x1,2 = 32.
Therefore, the vertices of the rectangle,
A(32, 22)B(-32, 22)C(-32, -22 and  D(32, -22).
The area of the rectangle A = 4 x  y = 4 (32 ) (22 ) = 48 square units.
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 33/2,
and intersections of the ellipse with the y-axis by setting  x = 0,   =>    9y2 -18y + 9 = 0,    y1,2 = 1.
Conic sections contents
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