document.write( "Question 1204640: Write the equation for the ellipse in standard form and general form.
\n" ); document.write( "foci at (-1,-1) and (9,-1), sum of focal radii 26 \n" ); document.write( "

Algebra.Com's Answer #841022 by ikleyn(52803) $\"\"$ $\"About$

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\n" ); document.write( "Write the equation for the ellipse in standard form and general form.
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document.write( "From given information about the foci coordinates, you can see that the \r\n" );
document.write( "distance between the foci points is 9 - (-1) = 10.\r\n" );
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document.write( "So, the half of this distance is 10/2 = 5 units.\r\n" );
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document.write( "This distance is the eccentricity, which has the standard designation \"c\".\r\n" );
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document.write( "So, c = 5 units.\r\n" );
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document.write( "Next, since the sum of focal radii is 26, it means that the distance from \r\n" );
document.write( "any focus of the ellipse to its any co-vertex is 26/2 = 13.\r\n" );
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document.write( "Thus we have a right angled triangle with one leg of 5 units and the hypotenuse of 13 units.\r\n" );
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document.write( "So, the other leg is   =  =  = 12 units.\r\n" );
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document.write( "Thus we found the minor semiaxis of the ellipse: it is 12 units.\r\n" );
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document.write( "The standard designation for the minor semi-axis of an ellipse is \"b\".\r\n" );
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document.write( "So, for our ellipse b = 12 units.\r\n" );
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document.write( "Finally, if \"a\" is the major semi-axis, then we have\r\n" );
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document.write( "     =  - ,\r\n" );
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document.write( "or\r\n" );
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document.write( "     =  - ,\r\n" );
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document.write( "     = 25 + 144 = 169\r\n" );
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document.write( "which implies\r\n" );
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document.write( "    a =  = 13.\r\n" );
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document.write( "Thus the major semi-axis is 13 units long.\r\n" );
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document.write( "Now the standard form of this ellipse equation is\r\n" );
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document.write( "     +  = 1.\r\n" );
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document.write( "It is because the center of the ellipse is at the point (4,-1).\r\n" );
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\n" ); document.write( "\n" ); document.write( "After that, to find the general equation is simple arithmetic.\r
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\n" ); document.write( "\n" ); document.write( "For basic info about ellipses, see the lesson\r
\n" ); document.write( "\n" ); document.write( " - Ellipse definition, canonical equation, characteristic points and elements \r
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