Question 120910
The endpoints of the major axis of an ellipse are (-4,-2) and (8,-2). The endpoints of the minor axis are (2,3) and (2,-7). Find the equation of this ellipse
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Plot those 4 points

{{{drawing(400,375,-6,10,-10,6,
graph(400,375,-6,10,-10,6), locate(-4.1,-1.5948,o), locate(7.9,-1.5948,o), locate(1.85,3.36,o), locate(1.85,-6.59,o)
 )}}}

Draw the major axis:

{{{drawing(400,375,-6,10,-10,6,
graph(400,375,-6,10,-10,6), line(-4,-2,8,-2),
locate(-4.1,-1.5948,o), locate(7.9,-1.5948,o), locate(1.85,3.36,o), locate(1.85,-6.59,o) )}}}

Notice that the major axis is 12 units long.
Therefore since the major axis is 2a units long,
2a = 12 and the semi-major axis, a = 6.

Draw the minor axis:

{{{drawing(400,375,-6,10,-10,6,
graph(400,375,-6,10,-10,6), line(-4,-2,8,-2), line(2,-7,2,3),
locate(-4.1,-1.5948,o), locate(7.9,-1.5948,o), locate(1.85,3.36,o), locate(1.85,-6.59,o) )}}}

Notice that the minor axis is 10 units long.
Therefore since the minor axis is 2b units long,
2b = 10 and the semi-minor axis, b = 5.

Notice that the major and minor axis cross at the
point (2, -2).  So that is the center of the
ellipse, so (h,k) = (2,-2)

Sketch in the ellipse:

{{{drawing(400,375,-6,10,-10,6,
graph(400,375,-6,10,-10,6,-2+sqrt(900-25(x-2)^2)/6),
graph(400,375,-6,10,-10,6,-2-sqrt(900-25(x-2)^2)/6), line(-4,-2,8,-2), line(2,-7,2,3),
locate(-4.1,-1.5948,o), locate(7.9,-1.5948,o), locate(1.85,3.36,o), locate(1.85,-6.59,o) )}}}   

The equation of an ellipse whose major axis 
is horizontal is

{{{(x-h)^2/a^2 + (y-k)^2/b^2}}} = {{{1}}}

We have observed fom the graph that a = 6,
b = 5, and (h, k) = (2,-2), so substituting, we have

{{{(x-2)^2/6^2 + (y-(-2))^2/5^2}}} = {{{1}}}

or

{{{(x-2)^2/36 + (y+2)^2/25}}} = {{{1}}}

Edwin</pre>