Question 604369: Write an equation of an ellipse with center at (0, 0), co-vertex at (-4 , 0), and focus at (0, 3).
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! O(0,0) is the center,
(-4,0), and of course, symmetrically (4,0) are the co-vertices,
so 4 is the semi-minor axis, b.
(0,3), and of course, symmetrically (0,-3) are the foci,
so 3 is the focal distance, c.
The vertices are at (0,a) and (0,-a), with a being the semi-major axis and
 , so 
and  so  or 
HOW I KNOW THAT  :
In the diagram below, O is the center of the ellipse, with axes XA and BY.
(I did not draw the whole ellipse, just the important points).
 A and X are vertices; B and Y covertices; C and Z foci.
The important segment lengths are:
OA=OX=a (the semi-major axis)
OB+OY=b (the semi-minor axis)
OC=OZ=c (the focal distance
Covertex B is one of the points of the ellipse.
The distances from B to focus C and to focus Z are the same BZ=BC.
Their sum BZ+BC=2BC is the same as the sum of distances to the foci for all points on the ellipse.
Vertex A is one of the points on the ellipse.
The sum of its distances to the foci is AC+AZ=AC+(AO+OZ)=(a-c)+(a+c)=2a
2BC=2a --> BC=a
In the right triangle OBC, Pythagoras theorem says that
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