document.write( "Question 1044354: Express the equation of the ellipse in standard form. Then,
\n" ); document.write( "give the coordinates of the center, vertex, the foci, and
\n" ); document.write( "the endpoints of the latus rectum. Draw the ellipse, its
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Algebra.Com's Answer #659809 by Edwin McCravy(20059) $\"\"$ $\"About$

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Question 1044336
\n" ); document.write( "Give the coordinates of the center, foci, and covertices
\n" ); document.write( "of the ellipse with equation \r
\n" ); document.write( "\n" ); document.write( "Express the equation of the ellipse in standard form.
\n" ); document.write( "Then, give the coordinates of the center, vertex, the
\n" ); document.write( "foci, and the endpoints of the latus rectum. Draw the
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\n" ); document.write( "\n" ); document.write( "3x²+7y²-12x-28y+19 = 0\r
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document.write( "Rearrange\r\n" );
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document.write( "Factor out coefficients of squared letters:\r\n" );
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document.write( "To complete the squares, we need to add a number to the \r\n" );
document.write( "end of each parentheses, and to the right side:\r\n" );
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document.write( "so we put a blank where we need to add numbers.\r\n" );
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document.write( "To complete the square in the first parentheses:\r\n" );
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document.write( "1.  Multiply the coefficient of x by :\r\n" );
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document.write( "2.  Square that result:\r\n" );
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document.write( "3.  Put that where the first blank is on the left side:\r\n" );
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document.write( "So we complete the square in the first parentheses by\r\n" );
document.write( "adding +4 inside the first parentheses\r\n" );
document.write( "which actually amounts to adding 3*4 or 12 to the left \r\n" );
document.write( "side because there is a 3 in front of the parentheses, so\r\n" );
document.write( "we must add 12 to the right side:\r\n" );
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document.write( "To complete the square in the second parentheses, since the \r\n" );
document.write( "coefficient of y in the second parentheses is the same as the \r\n" );
document.write( "coefficient of x in the first parentheses, we also put +4 in\r\n" );
document.write( "second blank on the left as well.\r\n" );
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document.write( "Since we complete the square in the second parentheses by adding +4 \r\n" );
document.write( "inside the second parentheses, that actually amounts to adding 7*4 \r\n" );
document.write( "or 28 to the left side because there is a 7 in front of the \r\n" );
document.write( "parentheses, so we must add 28 to the right side, so we put 28\r\n" );
document.write( "in the remaining blank on the right side:\r\n" );
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document.write( "We factor both parentheses as perfect squares of binomials,\r\n" );
document.write( "and combine the numbers on the right side:\r\n" );
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document.write( "Get a 1 on the right by dividing through by 21\r\n" );
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document.write( "Simplify.  \r\n" );
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document.write( "Since the largest denominator is under the term in\r\n" );
document.write( "x, the ellipse has a horizontal major axis.  So we\r\n" );
document.write( "compare it to:\r\n" );
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document.write( ", , \r\n" );
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document.write( " so \r\n" );
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document.write( " so \r\n" );
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document.write( "Its center is at (h,k) = (2,2)\r\n" );
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document.write( "Plot the center (2,2):\r\n" );
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document.write( "Draw the major axis  units both left and right\r\n" );
document.write( "of the center.\r\n" );
document.write( "Draw the minor axis  units both above and below\r\n" );
document.write( "the center.\r\n" );
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document.write( "The vertices are  units right and left the center (2,2)\r\n" );
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document.write( "So we add  to the x-coordinate of the center.  So the \r\n" );
document.write( "right vertex is \r\n" );
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document.write( "And we subtract  from the y-coordinate of the center.  So the \r\n" );
document.write( "left vertex is \r\n" );
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document.write( "The covertices are  units above and below the center (2,2)\r\n" );
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document.write( "So we add  to the y-coordinate of the center.  So the \r\n" );
document.write( "upper covertex is .\r\n" );
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document.write( "And we subtract  from the y-coordinate of the center.  \r\n" );
document.write( "So the lower covertex is .\r\n" );
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document.write( "Sketch in the ellipse:\r\n" );
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document.write( "To find the foci, we must calculate c, using the Pythagorean\r\n" );
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document.write( "The foci are  units above and below the \r\n" );
document.write( "center (2,2)\r\n" );
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document.write( "So we add  to the x-coordinate of the center\r\n" );
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document.write( "So the right focus is (4,2)\r\n" );
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document.write( "And we subtract  from the x-coordinate of \r\n" );
document.write( "the center.\r\n" );
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document.write( "So the left focus is (0,2)\r\n" );
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document.write( "We plot the two foci:\r\n" );
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document.write( "Finally we draw in the two latus rectums:\r\n" );
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document.write( "To find the endpoints of the two latus rectums, we substitute\r\n" );
document.write( "the x-coordinates of the foci in the original equation:\r\n" );
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document.write( "3x²+7y²-12x-28y+19 = 0\r\n" );
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document.write( "Substituting x=0, the x-coordinate of the left focus:\r\n" );
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document.write( "3(0)²+7y²-12(0)-28y+19 = 0\r\n" );
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document.write( "7y²-28y+19 = 0\r\n" );
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document.write( "So the endpoints of the left latus rectum are\r\n" );
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document.write( "By symmetry, the endpoints of the right latus rectum are\r\n" );
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document.write( "To find the directrices, we determine the eccentricity e\r\n" );
document.write( "by the formula \r\n" );
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document.write( "The distance from the focus to the endpoint of the latus rectum is \r\n" );
document.write( "So the above becomes:\r\n" );
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document.write( "Take reciprocals of both sides:\r\n" );
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document.write( "Multiply both sides by \r\n" );
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document.write( "To find the right directrix, we add  to the x-coordinate of\r\n" );
document.write( "the right focus and get  or  or 7.5.  So the\r\n" );
document.write( "right directrix is the vertical line whose equation is .\r\n" );
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document.write( "To find the left directrix, we subtract  to the x-coordinate of\r\n" );
document.write( "the left focus and get  or  or -3.5.  So the\r\n" );
document.write( "left directrix is the vertical line whose equation is .\r\n" );
document.write( "They are the blue lines below:\r\n" );
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document.write( "Edwin

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