document.write( "Question 1018042: Find the vertex and focus and end points of the latus rectum of the parabola 9x^2-12x-36y-8=0 \n" ); document.write( "
Algebra.Com's Answer #634312 by Edwin McCravy(20060)\"\" \"About 
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document.write( "\"9x%5E2-12x\"\"%22%22=%22%22\"\"36y%2B8\"\r\n" );
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document.write( "Divide through by 9\r\n" );
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document.write( "\"x%5E2-expr%2812%2F9%29x\"\"%22%22=%22%22\"\"4y%2B8%2F9\"\r\n" );
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document.write( "\"x%5E2-expr%284%2F3%29x\"\"%22%22=%22%22\"\"4y%2B8%2F9\"\r\n" );
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document.write( "Complete the square:\r\n" );
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document.write( "               \"expr%28-4%2F3%29%2Aexpr%281%2F2%29=-4%2F6=-2%2F3\"\r\n" );
document.write( "               \"%28-2%2F3%29%5E2=4%2F9\"\r\n" );
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document.write( "Add to both sides\r\n" );
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document.write( "\"x%5E2-expr%284%2F3%29x%2B4%2F9\"\"%22%22=%22%22\"\"4y%2B8%2F9%2B4%2F9\"\r\n" );
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document.write( "\"%28x-2%2F3%29%5E2\"\"%22%22=%22%22\"\"4y%2B12%2F9\"\r\n" );
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document.write( "\"%28x-2%2F3%29%5E2\"\"%22%22=%22%22\"\"4y%2B4%2F3\"\r\n" );
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document.write( "\"%28x-2%2F3%29%5E2\"\"%22%22=%22%22\"\"4%28y%2B1%2F3%29\"\r\n" );
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document.write( "\"%28x-h%29%5E2\"\"%22%22=%22%22\"\"4p%28y-k%29\"\r\n" );
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document.write( "h=2, 4p=4, p=1, k=-1/3\r\n" );
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document.write( "Vertex = (h,k) = \"%28matrix%281%2C3%2C2%2F3%2C%22%2C%22%2C-1%2F3%29%29\"\r\n" );
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document.write( "Length of latus rectum = 4p = 4.\r\n" );
document.write( "                          p = 1.\r\n" );
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document.write( "Distance from vertex to the focus = \r\n" );
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document.write( "Distance from vertex to the directrix = p = 1\r\n" );
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document.write( "y-coordinate of vertex = \"-1%2F3\", add p=1, get \"2%2F3\",\r\n" );
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document.write( "so focus = \"%28matrix%281%2C3%2C2%2F3%2C%22%2C%22%2C2%2F3%29%29\".\r\n" );
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document.write( "x-coordinate of right endpoint of latus rectum = \r\n" );
document.write( "x-ccordinate of focus plus half of latus rectum's length,\r\n" );
document.write( "2p, get \"2%2F3%2B2%281%29=2%2F3%2B6%2F3=8%2F3\"\r\n" );
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document.write( "x-coordinate of left endpoint of latus rectum = \r\n" );
document.write( "x-ccordinate of focus minus half of latus rectum's length,\r\n" );
document.write( "2p, get \"2%2F3-2%281%29=2%2F3-6%2F3=-4%2F3\"\r\n" );
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document.write( "y-coordinates of ends of latus rectum = same as y-coordinate\r\n" );
document.write( "of focus, \"2%2F3\"\r\n" );
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document.write( "right end of latus rectum = \"%28matrix%281%2C3%2C8%2F3%2C%22%2C%22%2C2%2F3%29%29\"\r\n" );
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document.write( "right end of latus rectum = \"%28matrix%281%2C3%2C-4%2F3%2C%22%2C%22%2C2%2F3%29%29\"\r\n" );
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document.write( "Directrix has equation y = y-coodinate of vertex minus p = 1\r\n" );
document.write( "which is \"-1%2F3-1=-1%2F3-3%2F3=-4%2F3\"\r\n" );
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document.write( "Directrix has equation \"y=-4%2F3\"\r\n" );
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document.write( "blue line segment is latus rectum\r\n" );
document.write( "green line is directrix\r\n" );
document.write( "Upper point marked is focus\r\n" );
document.write( "Lower point marked is vertex\r\n" );
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document.write( "Edwin
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