document.write( "Question 935933: Find the standard form of the equation of the ellipse with Major axis of length 8 and foci (5,1) and (-1,1) \n" ); document.write( "
Algebra.Com's Answer #569328 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The other tutors don't draw graphs.  But you can never \r\n" );
document.write( "learn conics if you don't draw graphs.  \r\n" );
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document.write( "Draw the foci which are given as (-1,1) and (5,1):\r\n" );
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document.write( "1. Plot the foci (-1,1) and (5,1).\r\n" );
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document.write( "This tells us that we have an ellipse like this \"drawing%2820%2C10%2C-2%2C2%2C-1%2C1%2Carc%280%2C0%2C-3.9%2C1.9%29+%29\", which has equation\r\n" );
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document.write( "\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1\"\r\n" );
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document.write( "The center (h,k) is the midpoint between the two foci which is (2,1),\r\n" );
document.write( "We plot that.\r\n" );
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document.write( "So we can fill in h and k in the equation h=2, k=1.\r\n" );
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document.write( "\"%28x-2%29%5E2%2Fa%5E2%2B%28y-1%29%5E2%2Fb%5E2=1\"\r\n" );
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document.write( "Now we draw the major axis which is 8 units long, with its center\r\n" );
document.write( "bisecting it, so it is 4 units to the right of the center and 4\r\n" );
document.write( "units left of the senter.  So we draw the major axis in green:\r\n" );
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document.write( "We know that a is the length of the semi-major axis or half of the major\r\n" );
document.write( "axis, so a=4, and we can fill in a=4 in the equation\r\n" );
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document.write( "\"%28x-2%29%5E2%2F4%5E2%2B%28y-1%29%5E2%2Fb%5E2=1\"\r\n" );
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document.write( "Ther vertices are the endpoints of the major axis, so they are (-2,1) and\r\n" );
document.write( "(6,1). We plot those:\r\n" );
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document.write( "Next we calculate b, the length of the semi-minor axis from the equation:\r\n" );
document.write( "\"c\" is the distance from the focus to the center, which is 3 units.\r\n" );
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document.write( "\"c%5E2=a%5E2-b%5E2\"\r\n" );
document.write( "\"3%5E2=4%5E2-b%5E2\"\r\n" );
document.write( "\"b%5E2=4%5E2-3%5E2\"\r\n" );
document.write( "\"b%5E2=16-9\"\r\n" );
document.write( "\"b%5E2=7\"\r\n" );
document.write( "\"b=sqrt%287%29\", about 2.65, and the center bisects the minor axis also.\r\n" );
document.write( "so we draw in the minor axis also, about 2.65 up from the center, and\r\n" );
document.write( "2.65 down from the center, like this:\r\n" );
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document.write( "And finally we can draw in the graph of the ellipse:\r\n" );
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document.write( "Since we now know that \"b=sqrt%287%29\", the equation is now:\r\n" );
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document.write( "\"%28x-2%29%5E2%2F4%5E2%2B%28y-1%29%5E2%2F%28sqrt%287%29%29%5E2=1\"\r\n" );
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document.write( "or\r\n" );
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document.write( "\"%28x-2%29%5E2%2F16%2B%28y-1%29%5E2%2F7=1\"\r\n" );
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document.write( "Edwin
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