document.write( "Question 325483: Find the center, foci, vertices, length of major axis, and length of minor axis of the ellipse 25(x-2)^2 + 4(y+5)^2 = 100. Sketch the graph of the ellipse. Thanks in advance... (^2 means squared) \n" ); document.write( "
Algebra.Com's Answer #233093 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The standard forms for ellipses are\r\n" );
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document.write( "\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2\"\"%22%22=%22%22\"\"1\" for ellipses like this \"drawing%2850%2C25%2C-10%2C10%2C-5%2C5%2C+arc%280%2C0%2C18%2C-8%29+++%29\" where a > b\r\n" );
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document.write( "\"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2\"\"%22%22=%22%22\"\"1\" for ellipses like this \"drawing%2825%2C50%2C-5%2C5%2C-10%2C10%2C+arc%280%2C0%2C8%2C-18%29+++%29\" where a < b\r\n" );
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document.write( "\"25%28x-2%29%5E2+%2B+4%28y%2B5%29%5E2\"\"%22%22=%22%22\"\"100\"\r\n" );
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document.write( "To get 1 on the right, divide through by \"100\"\r\n" );
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document.write( "\"25%28x-2%29%5E2%2F100+%2B+4%28y%2B5%29%5E2%2F100\"\"%22%22=%22%22\"\"100%2F100\"\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "\"%28x-2%29%5E2%2F4+%2B+%28y%2B5%29%5E2%2F25\"\"%22%22=%22%22\"\"1\"\r\n" );
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document.write( "Since the larger number is under the expression in y the ellipse is \r\n" );
document.write( "of the form \"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2\"\"%22%22=%22%22\"\"1\" \r\n" );
document.write( "and is like this \"drawing%2825%2C50%2C-5%2C5%2C-10%2C10%2C+arc%280%2C0%2C8%2C-18%29+++%29\" \r\n" );
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document.write( "\"h=2\", \"k=-5\", \"a%5E2=25\" or \"a=5\", \"b%5E2=4\", or \"b=2\"\r\n" );
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document.write( "center = (h,k) = (2,-5)\r\n" );
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document.write( "So let's begin by plotting the center (2,-5)\r\n" );
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document.write( "Next we draw a vertical line beginning at the center\r\n" );
document.write( "(2,-5) and going upward \"a=5\" units, which is one-half the\r\n" );
document.write( "major axis.  This ends in the point (2,0) which is the upper\r\n" );
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document.write( "Next we draw a vertical line beginning at the center\r\n" );
document.write( "(2,-5) and going downward \"a=5\" units, which is one-half the\r\n" );
document.write( "major axis.  This ends in the point (2,-10) which is the lower\r\n" );
document.write( "vertex.  That green line is the major axis, and it is 10 units long.\r\n" );
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document.write( "Next we draw a horizontal line beginning at the center\r\n" );
document.write( "(2,-5) and going to the right \"b=2\" units, which is one-half the\r\n" );
document.write( "minor axis. This ends in the point (4,-5) which is the right co-vertex. \r\n" );
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document.write( "Next we draw a horizontal line beginning at the center\r\n" );
document.write( "(2,-5) and going to the leftt \"b=2\" units, which is one-half the\r\n" );
document.write( "minor axis. This ends in the point (0,-5) which is the left co-vertex.\r\n" );
document.write( "The horizontal green line is is the minor axis, and it is 4 units long.\r\n" );
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document.write( "Now we can sketch in the ellipse:\r\n" );
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document.write( "Finally we find the foci.  They are the two points on the major axis\r\n" );
document.write( "w2hich are c units from the center, where c is calculated by\r\n" );
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document.write( "\"c%5E2=a%5E2-b%5E2\"\r\n" );
document.write( "\"c%5E2=5%5E2-2%5E2\"\r\n" );
document.write( "\"c%5E2=25-4\"\r\n" );
document.write( "\"c%5E2=21\"\r\n" );
document.write( "\"c=sqrt%2821%29\"\r\n" );
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document.write( "So we add \"sqrt%2821%29\" to the y-coordinate of the center\r\n" );
document.write( "to find the upper focus, which is the point\r\n" );
document.write( "(2,\"5%2Bsqrt%2821%29\"), which is about (2,9.6),\r\n" );
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document.write( "And we subtract \"sqrt%2821%29\" from the y-coordinate of the center\r\n" );
document.write( "to find the lower focus, which is the point\r\n" );
document.write( "(2,\"5-sqrt%2821%29\"), which is about (2,0.4),\r\n" );
document.write( "also marked in red below.\r\n" );
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document.write( "Edwin

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