Ellipse
      The parametric equations of the ellipse
      Equation of a translated ellipse
         Equations of the ellipse examples
 
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.
Equation of a translated ellipse -the ellipse with the center at (x0, y0) and the major axis parallel to the x-axis.
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.
Equations of the 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:  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 33/2,
and intersections of the ellipse with the y-axis by setting  x = 0,   =>    9y2 -18y + 9 = 0,    y1,2 = 1.
Pre-calculus contents H
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