document.write( "Question 874025: Q.there are exactly 2 points on the ellipse $\"+x%5E2%2Fa%5E2+%2B+y%5E2%2Fb%5E2=1+\"$ whose distance from the center of the ellipse are equal to $\"+sqrt%28a%5E2%2Bb%5E2%29%2Fsqrt%282%29+\"$ find the eccentricity of the ellipse.
\n" ); document.write( "here's what i tried:
\n" ); document.write( "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
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Algebra.Com's Answer #527404 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
There is a typo somewhere.
\n" ); document.write( "If an ellipse is given by the equation $\"x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=1\"$ , with $\"a%3E0\"$ and $\"b%3E0\"$ ,
\n" ); document.write( "and there is at least one point whose distance from the center of the ellipse is equal to
\n" ); document.write( " $\"sqrt%28a%5E2%2Bb%5E2%29%2Fsqrt%282%29\"$ ,
\n" ); document.write( "the ellipse is a circle, with radius $\"a=b=sqrt%28a%5E2%2Bb%5E2%29%2Fsqrt%282%29\"$ ,
\n" ); document.write( "and all the infinite number of points on the ellipse are at a distance of
\n" ); document.write( " $\"sqrt%28a%5E2%2Bb%5E2%29%2Fsqrt%282%29\"$ from the center of that ellipse, which is a circle.
\n" ); document.write( "
\n" ); document.write( "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,
\n" ); document.write( "with that distance as the radio, and centered at the center of the ellipse.
\n" ); document.write( "It could be that circle and ellipse share no points :
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.
\n" ); document.write( "It could be that circle and ellipse share exactly two points :
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.
\n" ); document.write( "It could be that circle and ellipse share exactly four points :
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,
\n" ); document.write( "or it could be that the ellipse IS the same curve as the circle, and so they share all their infinite number of points.
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\n" ); document.write( "The ellipse $\"x%5E2%2Fa%5E2+%2B+y%5E2%2Fb%5E2=1+\"$ , with $\"a%3E0\"$ and $\"b%3E0\"$ , is centered on the origin,
\n" ); document.write( "so we are dealing with an ellipse and a circle centered (both of them) at (0,0), the origin.
\n" ); document.write( "
\n" ); document.write( "The points the problem talks about are on the ellipse and on that circle.
\n" ); document.write( "If those points are exactly 2 points, they are either the vertices or the co-vertices.
\n" ); document.write( "That means, those points must be either (a,0) and {-a,0),
\n" ); document.write( "which are at a distance $\"a\"$ from the origin,
\n" ); document.write( "or those points must be either (0,b), and (0,-b),
\n" ); document.write( "which are at a distance $\"b\"$ from the origin.
\n" ); document.write( "
\n" ); document.write( "If the distance is $\"+sqrt%28a%5E2%2Bb%5E2%29%2Fsqrt%282%29+\"$ , as you posted,
\n" ); document.write( "then $\"a=sqrt%28a%5E2%2Bb%5E2%29%2Fsqrt%282%29+\"$ or $\"b=sqrt%28a%5E2%2Bb%5E2%29%2Fsqrt%282%29+\"$ .
\n" ); document.write( "However,
\n" ); document.write( " $\"a=sqrt%28a%5E2%2Bb%5E2%29%2Fsqrt%282%29\"$ --> $\"a%5E2=%28a%5E2%2Bb%5E2%29%2F%28sqrt%282%29%5E2%29\"$ --> $\"a%5E2=%28a%5E2%2Bb%5E2%29%2F2\"$ --> $\"2a%5E2=a%5E2%2Bb%5E2\"$ --> $\"a%5E2=b%5E2\"$ --> $\"a=b\"$ ,
\n" ); document.write( "and
\n" ); document.write( " $\"b=sqrt%28a%5E2%2Bb%5E2%29%2Fsqrt%282%29\"$ --> $\"b%5E2=%28a%5E2%2Bb%5E2%29%2F%28sqrt%282%29%5E2%29\"$ --> $\"b%5E2=%28a%5E2%2Bb%5E2%29%2F2\"$ --> $\"2b%5E2=a%5E2%2Bb%5E2\"$ --> $\"b%5E2=a%5E2\"$ --> $\"b=a\"$ .
\n" ); document.write( "Either way, the ellipse turns to be a circle of radius $\"a=b\"$ ,
\n" ); document.write( "and all of its infinite number of points are at a distance
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.
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\n" ); document.write( "NOTE - CHECK THE PROBLEM'S EQUATIONS:
\n" ); document.write( "If the distance were
\n" ); document.write( " $\"sqrt%28a%5E2%2Bb%5E2%29%2F2\"$ , then
\n" ); document.write( " $\"a=sqrt%28a%5E2%2Bb%5E2%29%2F2\"$ --> $\"a%5E2=%28a%5E2%2Bb%5E2%29%2F2%5E2\"$ --> $\"a%5E2=%28a%5E2%2Bb%5E2%29%2F4\"$ --> $\"4a%5E2=a%5E2%2Bb%5E2\"$ --> $\"3a%5E2=b%5E2\"$ ,
\n" ); document.write( "and if
\n" ); document.write( " $\"b=sqrt%28a%5E2%2Bb%5E2%29%2F2\"$ --> $\"b%5E2=%28a%5E2%2Bb%5E2%29%2F2%5E2\"$ --> $\"b%5E2=%28a%5E2%2Bb%5E2%29%2F4\"$ --> $\"4b%5E2=a%5E2%2Bb%5E2\"$ --> $\"3b%5E2=a%5E2\"$ .
\n" ); document.write( "
\n" ); document.write( "In either case, the eccentricity would be the same.
\n" ); document.write( "
\n" ); document.write( "If $\"3b%5E2=a%5E2\"$ <--> $\"3b%5E2=a%5E2\"$ , then $\"a%3Eb%3E0\"$ ;
\n" ); document.write( " $\"a=sqrt%283b%5E2%29=sqrt%283%29%2Ab\"$ is the semi-major axis;
\n" ); document.write( "the vertices are (a,0) and {-a,0);
\n" ); document.write( " $\"c%5E2=a%5E2%2Bb%5E2=3b%5E2%2Bb%5E2=4b%5E2\"$ --> c+sqrt(4b^2)=2b}}} ,
\n" ); document.write( "and the eccentricity is $\"c%2Fa=2b%2F%28sqrt%283%29%2Ab%29=2%2Fsqrt%283%29=2sqrt%283%29%2F3\"$ .
\n" ); document.write( "
\n" ); document.write( "If $\"3a%5E2=b%5E2\"$ <--> $\"b%5E2=3a%5E2\"$ , then $\"b%3Ea%3E0\"$ ;
\n" ); document.write( " $\"b=sqrt%283a%5E2%29=sqrt%283%29%2Aa\"$ is the semi-major axis;
\n" ); document.write( "the vertices are (0,b) and {0,-b);
\n" ); document.write( " $\"c%5E2=a%5E2%2Bb%5E2=a%5E2%2B3a%5E2=4a%5E2\"$ --> $\"c=sqrt%284a%5E2%29=2a\"$ ,
\n" ); document.write( "and the eccentricity is $\"c%2Fb=2a%2F%28sqrt%283%29%2Aa%29=2%2Fsqrt%283%29=2sqrt%283%29%2F3\"$ \n" ); document.write( "

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