document.write( "Question 923595: Find the equation of the parabola with vertex (-2,-1) & the equation of the directrix is x-2y+1=0? \n" ); document.write( "
Algebra.Com's Answer #560177 by Edwin McCravy(20059)\"\" \"About 
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Find the equation of the parabola with vertex (-2,-1) & the equation of the directrix is x-2y+1=0?
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document.write( "The black line is the given directrix.\r\n" );
document.write( "The black point is the given focus.\r\n" );
document.write( "The green line is the axis of symmetry, through the vertex \r\n" );
document.write( "perpendicular to the directrix.\r\n" );
document.write( "The red point is the focus, which we must find.\r\n" );
document.write( "The point P(x,y) is an arbitrary point on the parabola.\r\n" );
document.write( "By the definition of a parabola, the two blue lines (one from\r\n" );
document.write( "P(x,y) to the focus and the other from P(x,y) to the directrix)\r\n" );
document.write( "must be equal.\r\n" );
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document.write( "We find the perpendicular distance from the vertex to the directrix,\r\n" );
document.write( "(the length oif the left blue line) by using the formula:\r\n" );
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document.write( "The perpendicular distance from the point P(x1,y1)\r\n" );
document.write( "to the line Ax+y+C=0 is\r\n" );
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document.write( "d = \"abs%28Ax%5B1%5D%2BBy%5B1%5D%2BC%29%2Fsqrt%28A%5E2%2BB%5E2%29\"\r\n" );
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document.write( "That gives d = \"1%2Fsqrt%285%29\" or \"sqrt%285%29%2F5\".\r\n" );
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document.write( "Then we find the focus, which is on the axis of symmetry a distance\r\n" );
document.write( "of \"sqrt%285%29%2F5\" from the focus (or \"2sqrt%285%29%2F5\" from the directrix).\r\n" );
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document.write( "The focus turns out to be the point \"%28matrix%281%2C3%2C-9%2F5%2C%22%2C%22%2C-7%2F5%29%29\",\r\n" );
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document.write( "Next we pick an arbitrary point on the parabola P(x,y).\r\n" );
document.write( "We set its perpendicular distance to the directrix equal to its distance\r\n" );
document.write( "from the focus (we are setting the two blue lines equal in length):\r\n" );
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document.write( "\"abs%28x-2y%2B1%29%2Fsqrt%281%5E2%2B2%5E2%29\"\"%22%22=%22%22\"\"sqrt%28%28x%2B9%2F5%29%5E2%2B%28y%2B7%2F5%29%5E2%29%29\"\r\n" );
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document.write( "Simplify/square both sides and after much algebra you'll end up with the\r\n" );
document.write( "equation:\r\n" );
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document.write( "\"4x%5E2%2B4xy%2By%5E2%2B16x%2B18y%2B25+=+0\"\r\n" );
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document.write( "If you get stuck on any step, you can contact me through the thank-you\r\n" );
document.write( "note form below.  You can come back online to this problem any time and \r\n" );
document.write( "write something else in the thank-you note and I'll get back to you.\r\n" );
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document.write( "Edwin
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