document.write( "Question 1187463: Find the equation of the locus of a point which moves so that the sum of its distances from the points (2,1) and (8,1) is 10\r
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Algebra.Com's Answer #818437 by ikleyn(52800) $\"\"$ $\"About$

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document.write( "It is by the DEFINITION the locus of points such, that the sum of distances from two given points is constant, is an ELLIPSE.\r\n" );
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document.write( "So, the sough locus, or the sough set of points is an ellipse.\r\n" );
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document.write( "About this ellipse, we know that its major axis is horizontal  y = 1.\r\n" );
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document.write( "We also know that the center of this ellipse is the point  (5,1),  where 5 is the arithmetic mean of 2 and 8,\r\n" );
document.write( "which are x-coordinates of the foci.\r\n" );
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document.write( "We also know that the eccentricity of this ellipse is half the distance between the foci\r\n" );
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document.write( "    c =  =  = 3.\r\n" );
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document.write( "Finally, we can calculate the length of the major semi-axis \"a\".\r\n" );
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document.write( "For it, we use the fact thant the sum of distances from the foci to any point of the ellipse is 10.\r\n" );
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document.write( "In particular, the sum  (c+a) PLUS (a-c) from the foci to the right-most vertex of the ellipse is 10:\r\n" );
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document.write( "    (c+a) + (a-c) = 10.\r\n" );
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document.write( "It gives  2a = 10;  hence,  a = 10/2 = 5.\r\n" );
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document.write( "By knowing \"a\" and \"c\" of the ellipse, we easily find \"b\", the length of the minor semi-axis: it is\r\n" );
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document.write( "    b =  =  =  -  = 4.\r\n" );
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document.write( "Now we are in position to write the standard form equation of this ellipse.  It is\r\n" );
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document.write( "     +  = 1.\r\n" );
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\n" ); document.write( "\n" ); document.write( "As you see, on the way of solving this problem I applied some trick, by using the sum of distances to some special point.\r
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\n" ); document.write( "\n" ); document.write( "This trick is the major key point of the solution to this problem.\r
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\n" ); document.write( "\n" ); document.write( "For basic info about ellipses, see my lesson\r
\n" ); document.write( "\n" ); document.write( " - Ellipse definition, canonical equation, characteristic points and elements \r
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