Question 874025: Q.there are exactly 2 points on the ellipse  whose distance from the center of the ellipse are equal to  find the eccentricity of the ellipse.
here's what i tried:
Ans)since the ellipse is symmetric about x and y axis if there are only 2 points which satisfy a condition, they must lie on the x and y axis. which means that the two points are extremities of the ellipse on y-axis. thus on solving given distance = 2b we get e=sqrt(6/7) which is not the right answer
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! There is a typo somewhere.
If an ellipse is given by the equation  , with  and  ,
and there is at least one point whose distance from the center of the ellipse is equal to
 ,
the ellipse is a circle, with radius  ,
and all the infinite number of points on the ellipse are at a distance of
from the center of that ellipse, which is a circle.
The points on an ellipse whose distance from the center of the ellipse are equal to a given distance are all the points of a circle,
with that distance as the radio, and centered at the center of the ellipse.
It could be that circle and ellipse share no points :
 or  .
It could be that circle and ellipse share exactly two points :
 or  .
It could be that circle and ellipse share exactly four points :
 ,
or it could be that the ellipse IS the same curve as the circle, and so they share all their infinite number of points.
The ellipse  , with and , is centered on the origin,
so we are dealing with an ellipse and a circle centered (both of them) at (0,0), the origin.
The points the problem talks about are on the ellipse and on that circle.
If those points are exactly 2 points, they are either the vertices or the co-vertices.
That means, those points must be either (a,0) and {-a,0),
which are at a distance  from the origin,
or those points must be either (0,b), and (0,-b),
which are at a distance  from the origin.
If the distance is , as you posted,
then  or  .
However,
 --> --> --> --> --> ,
and
 --> --> --> --> --> .
Either way, the ellipse turns to be a circle of radius ,
and all of its infinite number of points are at a distance
 .
NOTE - CHECK THE PROBLEM'S EQUATIONS:
If the distance were
 , then
 --> --> --> --> ,
and if
 --> --> --> --> .
In either case, the eccentricity would be the same.
If <--> , then  ;
 is the semi-major axis;
the vertices are (a,0) and {-a,0);
 --> c+sqrt(4b^2)=2b}}} ,
and the eccentricity is  .
If <--> , then  ;
 is the semi-major axis;
the vertices are (0,b) and {0,-b);
 -->  ,
and the eccentricity is 
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