document.write( "Question 675159: Let f(x) be the parabola -x^2+16x-16. Find the point (x,y) on f such that tangent line to f at (x,y) passes through the origin. \n" ); document.write( "

Algebra.Com's Answer #419649 by Edwin McCravy(20060) $\"\"$ $\"About$

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Let f(x) be the parabola -x^2+16x-16. Find the point (x,y) on f such that tangent line to f at (x,y) passes through the origin.
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document.write( "The way to approach this depends on whether you are taking algebra or\r\n" );
document.write( "taking calculus.  I will assume you are taking algebra.  If you are\r\n" );
document.write( "taking calculus, tell me in the thank you note and I'll help you\r\n" );
document.write( "using calculus.\r\n" );
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document.write( "If you are taking algebra.  You need this fact:\r\n" );
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document.write( "The graph of a line g(x) is tangent to the graph of parabola f(x) \r\n" );
document.write( "if and only if the equation g(x)=f(x) has exactly one solution.\r\n" );
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document.write( "Let the line that is to be tangent to the parabola f(x) have the \r\n" );
document.write( "equation g(x) = mx + b.  Since g(x) is to pass through the origin\r\n" );
document.write( "(x,y) = (0,0) must satisfy that equation:\r\n" );
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document.write( "              g(x) = mx + b\r\n" );
document.write( "                 0 = m·0 + b\r\n" );
document.write( "                 0 = b\r\n" );
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document.write( "So            g(x) = mx + 0\r\n" );
document.write( "              g(x) = mx\r\n" );
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document.write( "Now we form the equation:\r\n" );
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document.write( "              g(x) = f(x)\r\n" );
document.write( "                mx = -x² + 16x - 16\r\n" );
document.write( "x² + mx - 16x + 16 = 0\r\n" );
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document.write( " x² + (m-16)x + 16 = 0\r\n" );
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document.write( "That equation will have exactly one solution if its\r\n" );
document.write( "discriminant B²-4AC = 0, where A=1, B = m-16, C=16\r\n" );
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document.write( "discriminant = B²-4AC = (m-16)²-4(1)(16) = (m-16)²-64\r\n" );
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document.write( "We set that discriminant = 0\r\n" );
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document.write( "     (m-16)² - 64 = 0\r\n" );
document.write( "          (m-16)² = 64\r\n" );
document.write( "Use the principle of square roots:\r\n" );
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document.write( "             m-16 = ±8\r\n" );
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document.write( "                m = 16±8\r\n" );
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document.write( "There are two solutions for m, 16+8=24 and 16-8=8\r\n" );
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document.write( "Using m = 24\r\n" );
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document.write( " x² + (m-16)x + 16 = 0 \r\n" );
document.write( " x² + (24-16)x + 16 = 0\r\n" );
document.write( "       x² + 8x + 16 = 0\r\n" );
document.write( "           (x + 4)² = 0\r\n" );
document.write( "                  x = -4\r\n" );
document.write( "The line's equation is g(x) = mx\r\n" );
document.write( "                       g(x) = 24x\r\n" );
document.write( "                      g(-4) = 24(-4)\r\n" );
document.write( "                      g(-4) = -96 \r\n" );
document.write( "So the point of tangency is (-4,-96)\r\n" );
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document.write( "The graph is:\r\n" );
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document.write( "There is another solution:\r\n" );
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document.write( "Using m = 8\r\n" );
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document.write( " x² + (m-16)x + 16 = 0 \r\n" );
document.write( " x² + (8-16)x + 16 = 0\r\n" );
document.write( "       x² - 8x + 16 = 0\r\n" );
document.write( "           (x - 4)² = 0\r\n" );
document.write( "                  x = 4\r\n" );
document.write( "The line's equation is g(x) = mx\r\n" );
document.write( "                       g(x) = 8x\r\n" );
document.write( "                      g(4) = 8(4)\r\n" );
document.write( "                      g(4) = 32 \r\n" );
document.write( "So the point of tangency is (4,32)\r\n" );
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document.write( "The graph is:\r\n" );
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document.write( "Edwin

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