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Question 605622: locate the center, foci, vertices, and ends of the latera recta of the ellipse. find the equation of the ellipse satisfying the given conditions. a focus at (-3,-1), one end of the minor axis at (0,3), major axis vertical
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! The major axis is vertical and passes through focus (-3,-1), so it is the line x=-3.
The minor axis has to be horizontal, and since it ends at (0,3), it must be the line y=3.
Now I know that the center of the ellipse is at (-3,3), where the minor and major axes intercept.
The length of the semi-minor axis is the distance between center (-3,3) and minor axis end (0,3):
 --> 
The focal distance is the distance between center (-3,3) and focus (-3,-1):
 --> 
The length of the semi-major axis of an ellipse can be calculated as
 so  --> 
With that we can write the equation as  (with  under  because the major axis is parallel top the y-axis)
 -->  .
The value of  also gives us the position of the vertices (the ends of the major axis), which are at a distance 5 above and below the center: (-3,8) and (-3, -2).
The other end of the minor axis is at a distance  from the center and on the other side, at (-6,3) and the other focus is at a distance  from the center and on the other side, at (-3,7).
The length of the semi-latus rectum can be calculated as
 so in this case  and the latus rectum is  .
The semi-latus rectum gives us the horizontal distance from the foci to ellipse points on either side of them (the 4 ends of the latera recta, with y=-1 or y=7 and
 and  .
We can plot all the meaningful points:
and then we could even graph the ellipse:
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