document.write( "Question 1191710: Derive the equation of the locus of a point P(x,y) which moves so that its distance from (2,3) is always equal to its distance from the line x+2=0 \n" ); document.write( "

Algebra.Com's Answer #823533 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Derive the equation of the locus of a point P(x,y) which moves so that
\n" ); document.write( "its distance from (2,3) is always equal to its distance from the line x+2=0.
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document.write( "The line x+2 = 0 is the line  x= -2  (vertical line parallel to y-axis with x-coordinate of -2).\r\n" );
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document.write( "Let (x,y) be the point of the locus.  Then the distance from (x,y) to the line x= -2 is  |x+2|\r\n" );
document.write( "(notice the absolute value sign).\r\n" );
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document.write( "The distance from (x,y) to the point (2,3)  is  .\r\n" );
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document.write( "The equation of the locus is\r\n" );
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document.write( "    = |x+2|.\r\n" );
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document.write( "Square both sides and get\r\n" );
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document.write( "    (x-2)^2 + (y-3)^2 = (x+2)^2\r\n" );
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document.write( "    x^2 - 4x + 4 + y^2 - 6y + 9 = x^2 + 4x + 4\r\n" );
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document.write( "    y^2 - 6y + 9 = 8x\r\n" );
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document.write( "    (y-3)^2 = 8x\r\n" );
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document.write( "It is final equation of the locus.  It represents a parabola with the horizontal axis y= 3, \r\n" );
document.write( "parallel to x-axis. The parabola is opened right. Its vertex is the point (x,y) = (0,3).\r\n" );
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