document.write( "Question 1181254: Given points A(0, 1), B(2, 5) and point P moving along the parabola y = x^2 + 4x + 7 …eq. 1, find the minimum value of area S of triangle PAB \n" ); document.write( "

Algebra.Com's Answer #811120 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "Points A and B determine the line with equation y=2x+1.

\n" ); document.write( "Here is a graph of that line and the given quadratic:

\n" ); document.write( " $\"graph%28400%2C400%2C-5%2C5%2C-1%2C10%2C2x%2B1%2Cx%5E2%2B4x%2B7%29\"$

\n" ); document.write( "Using AB as the base of the triangle, with P somewhere on the parabola, the minimum area of triangle PAB will be when the height of the triangle is smallest; that will happen when P is at the point on the parabola which is closest to the line. And that will be where the tangent to the parabola has the same slope as the line.

\n" ); document.write( "y=x^2+4x+7
\n" ); document.write( "y'=2x+4

\n" ); document.write( "The slope of the line is 2; find the point on the parabola where the slope of the tangent is 2:

\n" ); document.write( "2x+4=2
\n" ); document.write( "2x=-2
\n" ); document.write( "x=-1
\n" ); document.write( "y=1-4+7=4

\n" ); document.write( "The area of PAB is minimum when P is (-1,4).

\n" ); document.write( "Then use the \"shoelace\" method to find the area of the triangle with vertices (0,1), (2,5), and (-1,4).

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document.write( "\n" );
document.write( "      0   1
\n" );
document.write( "        X
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document.write( "      2   5
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document.write( "    /   X   \
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document.write( "  2  -1   4   0
\n" );
document.write( "    /   X   \
\n" );
document.write( " -5   0   1   8
\n" );
document.write( "    /       \
\n" );
document.write( "  0          -1
\n" );
document.write( " ---         ---
\n" );
document.write( " -3           7\r
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document.write( "\n" );
document.write( "Area = (1/2)(7-(-3)) = 5

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document.write( "ANSWER: The minimum area of PAB is 5.

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