document.write( "Question 1114072: What are the steps to find the equation of an ellipse with foci (0,-3) and (0,3); the point (8,3) on the ellipse? \n" ); document.write( "

Algebra.Com's Answer #729083 by ikleyn(52803) $\"\"$ $\"About$

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document.write( "1.  First of all, you see that the foci lie in vertical line x= 0 and the center of the ellipse, \r\n" );
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document.write( "    which is the midpoint between the foci,  coincides with the origin of the coordinate system  (0,0).\r\n" );
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document.write( "    So, the ellipse is centered at (0,0).\r\n" );
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document.write( "2.  Let  F1 = (0,-3)  and  F2 = (0,3)  be the names of the foci.\r\n" );
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document.write( "    The distance from F1 to the point (8,3) at the ellipse is   =  = 10 units.\r\n" );
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document.write( "    The distance from F2 to the point (8,3) at the ellipse is   =  = 8 units.\r\n" );
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document.write( "    The sum of these distances is 18 units.  \r\n" );
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document.write( "    According to the definition of an ellipse,  the sum of distances from foci to any point on the ellipse is constant.\r\n" );
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document.write( "    So, for our ellipse the sum of distances from foci to any points on the ellipse is 18 units.\r\n" );
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document.write( "3.  Let \"a\" and \"b\" be semi-major and semi-minor axes of our ellipse, respectively \r\n" );
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document.write( "    (a\" is along the vertical axis and \"b\" is along the horizontal axis).   \r\n" );
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document.write( "    At given conditions, the point B = (b,0) lies at the ellipse;  indeed, it is the co-vertice point of the ellipse.\r\n" );
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document.write( "    Due to symmetry, the distances from F1 and F2 to the point B are the same, and each of the distances is half of 18, i.e. 9 units.\r\n" );
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document.write( "    So the three points F1, (0,0) and B are vertices of the right-angled triangle with the hypotenuse of 9 units long and one of the leg 3 units long\r\n" );
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document.write( "    (from F1 to the center of the ellipse).\r\n" );
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document.write( "    Hence, the other leg is   =  = .\r\n" );
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document.write( "    This leg is nothing else as the minor semi-axis b.  Thus  b = .\r\n" );
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document.write( "4.  Now for our ellipse we know the linear eccentricity  c = 3 units  and  the minor semi-axis b = .\r\n" );
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document.write( "    Then the major semi-axis  a =  =  =  =  = 9.\r\n" );
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document.write( "5.   Having this, we can write the equation of our ellipse:\r\n" );
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document.write( "      +  = 1,   or     +  = 1.     (1)\r\n" );
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\n" ); document.write( "\n" ); document.write( "You may check that the given point (8,3) belongs to the ellipse, i.e. satisfies equation (1).\r
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