document.write( "Question 1181253: Given Point P moving along curve y = x^2 - 1 and point Q moving along line y = x - 3, find the coordinates of point Q and the distance between P and Q when the distance between those two points is minimized. \n" ); document.write( "

Algebra.Com's Answer #811111 by ikleyn(52787) $\"\"$ $\"About$

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\n" ); document.write( "Given Point P moving along curve y = x^2 - 1 and point Q moving along line y = x - 3,
\n" ); document.write( "find the coordinates of point Q and the distance between P and Q
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\n" ); document.write( "\n" ); document.write( "            My common sense tells me that we should find the point on the parabola, where the tangent line
\n" ); document.write( "            to the parabola is parallel to the given line, i.e. has the same slope of 1 as the given straight line.\r
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\n" ); document.write( "\n" ); document.write( "            Then the sough distance is the distance from this point to the line.\r
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document.write( "To find the point on the parabola with the slope of the tangent line equal to one,\r\n" );
document.write( "take the derivative and equate it to 1\r\n" );
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document.write( "     = 2x = 1,  which gives  x= 0.5, y= 0.5^2 - 3 = 0.25 - 3 = -2.75.\r\n" );
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document.write( "To find the distance from the point (0.5,-2.75) to the line y = x-3, use the general formula for the distance\r\n" );
document.write( "from the point (p,q) in the coordinate plane to the straight line Ax + By + C = 0.\r\n" );
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document.write( "This formula is  d = .\r\n" );
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document.write( "In our case,  the equation of the line is x - y - 3 = 0,  so  A = 1,  B = -1, C = -3,  p= 0.5,  q= -2.75.\r\n" );
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document.write( "We substitute these values into the formula (*) and get the answer\r\n" );
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document.write( "    the distance  d =  =  =  =  =  = 0.17678 (rounded).\r\n" );
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document.write( "ANSWER.  The minimal distance is   = 0.17678 (rounded).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "Do not forget to post your \"THANKS\" to me for my teaching.\r
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