document.write( "Question 689945: Find the equation of tangent to the hyperbola x^2-y^2=16 passing through the points(-1,-7) \n" ); document.write( "
Algebra.Com's Answer #426153 by Edwin McCravy(20059)\"\" \"About 
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document.write( "Using calculus would make things easier, but I will assume you know\r\n" );
document.write( "only algebra.\r\n" );
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document.write( "We use two facts:\r\n" );
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document.write( "1. A line is tangent to a curve if the system consisting \r\n" );
document.write( "of the equations of the line and the conic section has exactly one \r\n" );
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document.write( "2. A quadratic equation has exactly one solution if the\r\n" );
document.write( "discriminant B²-4AC = 0\r\n" );
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document.write( "Let the slope of the tangent line be m.\r\n" );
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document.write( "Since the line passes through (-1,-7), we use the point-slope form:\r\n" );
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document.write( "y-y1 = m(x-x1)\r\n" );
document.write( "y-(-7) = m(x-(-1))\r\n" );
document.write( "y+7 = m(x+1)\r\n" );
document.write( "y+7 = mx+m\r\n" );
document.write( "  y = mx+m-7    <---(The equation of the tangent line)\r\n" );
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document.write( "We know that if we solve this system\r\n" );
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document.write( "\"system%28++y+=+mx%2Bm-7%2Cx%5E2-y%5E2=16%29\"\r\n" );
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document.write( "we would get (x,y) = the point of tangency,\r\n" );
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document.write( "So we solve that by substitution:\r\n" );
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document.write( "x² - (mx+m-7)² = 16\r\n" );
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document.write( "x² - (m²x²+m²+49+2m²x-14mx-14m) = 16\r\n" );
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document.write( "x² - m²x² - m² - 49 - 2m²x + 14mx + 14m = 16\r\n" );
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document.write( "x² - m²x² - 2m²x + 14mx - m² - 49 + 14m = 16\r\n" );
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document.write( "(1-m²)x² + (-2m²+14m)x + (-m²+14m-65) = 0\r\n" );
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document.write( "This must have a single solution, so its discriminant\r\n" );
document.write( "B²-4AC = 0\r\n" );
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document.write( "Discriminant = (-2m²+14m)² - 4(1-m²)(-m²+14m-65) =\r\n" );
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document.write( "(4m4-56m³+196m²) - (4-4m²)(-m²+14m-65) =\r\n" );
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document.write( "(4m4-56m³+196m²) - (-4m²+56m-260+4m4-56m³+260m²) =\r\n" );
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document.write( "4m4 - 56m³ + 196m² + 4m² - 56m + 260 - 4m4 + 56m³ - 260m² =\r\n" );
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document.write( "-60m² - 56m + 260\r\n" );
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document.write( "This must = 0 so that the line will be tangent to the\r\n" );
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document.write( "hyperbola:\r\n" );
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document.write( "-60m² - 56m + 260 = 0\r\n" );
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document.write( "Divide through by -4\r\n" );
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document.write( "15m² + 14m - 65 = 0\r\n" );
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document.write( "(3m-5)(5m+13) = 0\r\n" );
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document.write( "3m-5 = 0;  5m+13 = 0\r\n" );
document.write( "  3m = 5;     5m = -13\r\n" );
document.write( "   m = \"5%2F3\";   m = \"-13%2F5\"\r\n" );
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document.write( "So we have two solutions:\r\n" );
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document.write( "using m = \"5%2F3\"    \r\n" );
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document.write( "  y = mx+m-7 becomes\r\n" );
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document.write( "  y = \"5%2F3\"x + \"5%2F3\" - 7\r\n" );
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document.write( "Multiply through by 3\r\n" );
document.write( "      \r\n" );
document.write( " 3y = 5x + 5 - 21\r\n" );
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document.write( " 3y = 5x - 16\r\n" );
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document.write( "-5x + 3y = -16\r\n" );
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document.write( " 5x - 3y = 16\r\n" );
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document.write( "using m = \"-13%2F5\"    \r\n" );
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document.write( "  y = mx+m-7 becomes\r\n" );
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document.write( "  y = \"-13%2F5\"x + \"-13%2F5\" - 7\r\n" );
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document.write( "Multiply through by 3\r\n" );
document.write( "      \r\n" );
document.write( " 5y = -13x - 13 - 35\r\n" );
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document.write( " 5y = -13x - 48\r\n" );
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document.write( "13x + 5y = -48\r\n" );
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document.write( "So there are two tangent lines.  Their equations are\r\n" );
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document.write( " 5x - 3y = 16  and  13x + 5y = -48.\r\n" );
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document.write( "Drawing in the other one: \r\n" );
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document.write( "Edwin
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