document.write( "Question 1170467: The foci of an ellipse are (-3,-6) and (-3,2). For any point on the ellipse, the
\n" ); document.write( "sum of its distances from the foci is 14. Find the standard equation of the
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Algebra.Com's Answer #795341 by ikleyn(52810) $\"\"$ $\"About$

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document.write( "From the condition, the major semi-axis is vertical and the distance between the foci is  2 - (-6) = 8.\r\n" );
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document.write( "Therefore, the eccentricity  \"e\"  is half of this distance, i.e.  e = 8/2 = 4.\r\n" );
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document.write( "Let the foci be the points F1 and F2, and let V be the vertex of the ellipse closest to F1.\r\n" );
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document.write( "Then the distance from  F1  to  V  is (a-e),  while the distance from F2 to V is (a+e), where \"a\" is the major semi-axis.\r\n" );
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document.write( "The sum of these distances is equal to 14, according to the condition\r\n" );
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document.write( "    (a-e) + (a+e) = 14,\r\n" );
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document.write( "which implies\r\n" );
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document.write( "    2a = 14,  a = 7.\r\n" );
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document.write( "Thus the major semi-axis is a = 7 and the eccentricity is 4.\r\n" );
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document.write( "Then  from  c = ,  we have for the minor semi-axis \"b\"\r\n" );
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document.write( "     = ,\r\n" );
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document.write( "     =  =  = 49 - 16 = 33.\r\n" );
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document.write( "The center of the ellipse is at  (-3,-2);  therefore, its equation is\r\n" );
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document.write( "     +  = 1.\r\n" );
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