document.write( "Question 1054909: A parabola has an equation of y^2 = 8x. Find the equation of
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Algebra.Com's Answer #670109 by Edwin McCravy(20056)\"\" \"About 
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A parabola has an equation of y^2 = 8x. Find the equation of
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document.write( "A diameter of a parabola is a ray from a point on the\r\n" );
document.write( "parabola parallel to the axis of symmetry of the parabola.\r\n" );
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document.write( "The parabola above is the graph of y2 = 8x.\r\n" );
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document.write( "The black line is the graph of x - y = 4.  \r\n" );
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document.write( "The red, green, and blue line segments are examples of \r\n" );
document.write( "chords parallel to the black line.  The red dotted line \r\n" );
document.write( "is the required diameter of the parabola which we will\r\n" );
document.write( "show bisects all chords parallel to the black line. \r\n" );
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document.write( "Every line parallel to x - y = 4, or y = x - 4, has slope \r\n" );
document.write( "(or gradient) = 1.\r\n" );
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document.write( "Let y = x + b be the equation of any chord of the parabola. Then\r\n" );
document.write( "we solve the system:\r\n" );
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document.write( "\"system%28y%5E2=8x%2Cy=x%2Bb%29\"\r\n" );
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document.write( "by substitution and we find that the coordinates of \r\n" );
document.write( "the endpoints of the chord are \r\n" );
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document.write( "\"%28matrix%281%2C3%2C4-b-2sqrt%284-2b%29%2C%22%2C%22%2C4-2sqrt%284-2b%29%29%29\"\r\n" );
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document.write( "and\r\n" );
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document.write( "\"%28matrix%281%2C3%2C4-b%2B2sqrt%284-2b%29%2C%22%2C%22%2C4%2B2sqrt%284-2b%29%29%29\"\r\n" );
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document.write( "Then we find the coordinates of the midpoint of the\r\n" );
document.write( "chord, using the midpoint formula\r\n" );
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document.write( "Midpoint = \r\n" );
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document.write( "Midpoint = \r\n" );
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document.write( "which simplifies to the point (4-b, 4).\r\n" );
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document.write( "So all the y-coordinates of all the midpoints of all chords \r\n" );
document.write( "parallel to the black line are all 4, so they are all on the line \r\n" );
document.write( "y = 4.\r\n" );
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document.write( "This is therefore the equation of the red dotted line which\r\n" );
document.write( "we are looking for.\r\n" );
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document.write( "Answer: the equation of the diameter of the parabola which\r\n" );
document.write( "bisects all the chords of the parabola parallel to the line\r\n" );
document.write( "x-y = 4 is:\r\n" );
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document.write( "    y = 4\r\n" );
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document.write( "Edwin
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