Conic Sections
    Ellipse
      Equation of a translated ellipse
         Ellipse examples
Equation of a translated 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.
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:   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.
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.
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