document.write( "Question 810701: Find the equation of the parabola whose focus is (-3, 2) and directrix is x+y = 4. \n" ); document.write( "
Algebra.Com's Answer #488377 by Edwin McCravy(20056)\"\" \"About 
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document.write( "Find the equation of the parabola whose focus is (-3, 2) and directrix is x+y = 4.\r\n" );
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document.write( "By definition a parabola is the set of points each of which is the same\r\n" );
document.write( "distance from a given point (called the focus) as its perpendicular \r\n" );
document.write( "distance to a given line (called the directrix).\r\n" );
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document.write( "Let (x,y) be any arbitrary point on the parabola.\r\n" );
document.write( "The green line is the directrix with equation x+y = 4.\r\n" );
document.write( "Draw two lines, one from that arbitrary point (x,y) \r\n" );
document.write( "perpendicular to the directrix and another from (x,y)\r\n" );
document.write( "to the focus (-3,2): \r\n" );
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document.write( "Those two black lines must be equal in length.\r\n" );
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document.write( "The perpendicular distance from the point (x1,y1)\r\n" );
document.write( "to the line Ax+By+C=0 is\r\n" );
document.write( "d = \"abs%28%28Ax%5B1%5D%2BBy%5B1%5D%2BC%29%2Fsqrt%28A%5E2%2BB%5E2%29%29\"\r\n" );
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document.write( "Therefore the perpendicular distance from any arbitrary point \r\n" );
document.write( "(x,y) on the desired parabola to the line x+y=4 or x+y-4=0 is\r\n" );
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document.write( "d = \"abs%28%281x%2B1y-4%29%2Fsqrt%281%5E2%2B1%5E2%29%29\" = \"abs%28x%2By-4%29%2Fsqrt%282%29%29\" = \"abs%28%28x%2By-4%29%2Fsqrt%281%2B1%29%29\" = \"abs%28x%2By-4%29%2Fsqrt%282%29%29\"\r\n" );
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document.write( "The distance between two points (x1,y1) and (x2,y2)\r\n" );
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document.write( "is given by the formula \r\n" );
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document.write( "d = √(x2-x1)²+(y2-y1)²\r\n" );
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document.write( "So the distance from any arbitrary point (x,y) and (-3,2) is given by:\r\n" );
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document.write( "d = √(-3-x)²+(2-y)²\r\n" );
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document.write( "So we set the two distances equal:\r\n" );
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document.write( "\"abs%28x%2By-4%29%2Fsqrt%282%29%29\" = √(-3-x)²+(2-y)²\r\n" );
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document.write( "Multiply both sides by √2 \r\n" );
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document.write( "|x+y-4| = √2√(-3-x)²+(2-y)²\r\n" );
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document.write( "Squaring both sides gets rid of the absolute value and the square roots:\r\n" );
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document.write( "(x+y-4)² = 2[(-3-x)²+(2-y)²]\r\n" );
document.write( "x²+y²+16+2xy-8x-8y = 2[9+6x+x²+4-4y+y²]\r\n" );
document.write( "x²+y²+16+2xy-8x-8y = 18+12x+2x²+8-8y+2y²\r\n" );
document.write( "x²+y²+16+2xy-8x-8y = 26+12x+2x²-8y+2y²\r\n" );
document.write( " -x²+2xy-y²-20x-10 = 0\r\n" );
document.write( "  x²-2xy+y²+20x+10 = 0\r\n" );
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document.write( "Edwin
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