document.write( "Question 1189323: Find the tangent line to the parabola x^2 = 6y + 10 through (7,5)
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Algebra.Com's Answer #820658 by ikleyn(52803) $\"\"$ $\"About$

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\n" ); document.write( "Find the tangent lines to the parabola x^2 = 6y + 10 passing through the point (7,5), which lies outside the parabola.
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\n" ); document.write( "\n" ); document.write( "            The final equations of the tangent lines are correct in the post by @Alan,\r
\n" ); document.write( "\n" ); document.write( "            but the logic of his solution is unclear to me.\r
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\n" ); document.write( "\n" ); document.write( "            I tried to understand it, but failed.\r
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\n" ); document.write( "\n" ); document.write( "            Therefore, I developed my own solution in a way as it SHOULD be done: \r
\n" ); document.write( "\n" ); document.write( "            in a way as my teachers and my textbooks taught me.\r
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document.write( "An equation of the line passing through the point (7,5) is\r\n" );
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document.write( "    y-5 = m*(x-7),\r\n" );
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document.write( "where \"m\" is the slope coefficient, or\r\n" );
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document.write( "    y = mx - 7m +5.\r\n" );
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document.write( "    +----------------------------------------------------------------+\r\n" );
document.write( "    |        The value of the slope \"m\" is unknown now,              |\r\n" );
document.write( "    |    and the rest of the solution is to find the value of \"m\".   |\r\n" );
document.write( "    +-----------------------------------------------------------==---+\r\n" );
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document.write( "Substitute this expression for y into the right side of the parabola formula\r\n" );
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document.write( "    x^2 = 6(mx - 7m +5) + 10.\r\n" );
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document.write( "You will get\r\n" );
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document.write( "    x^2 - 6mx + 42m - 40 = 0.    (1)\r\n" );
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document.write( "The line is tangent to the parabola if and only if the quadratic equation (1) has a unique real root.\r\n" );
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document.write( "It happens if and only if the discriminant of the equation is zero.\r\n" );
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document.write( "The discriminant is  d = b^2 - 4ac;  b= -6m;  a= 1;  c= 42m-40,  so\r\n" );
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document.write( "    d = 36m^2 - 4(42m-40) = 36m^2 - 168m + 160.\r\n" );
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document.write( "Therefore, to find \"m\", we need solve this quadratic equation\r\n" );
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document.write( "    36m^2 - 168m + 160 = 0.\r\n" );
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document.write( "Find its solutions, using the quadratic formula. They are\r\n" );
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document.write( "     =   and   = .\r\n" );
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document.write( "So, there are two tangent line through the given point to parabola.\r\n" );
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document.write( "Their equations are  y-5 =   and  y-5 = .\r\n" );
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document.write( "The equivalent forms of these equations are\r\n" );
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document.write( "    3y - 10x = -55  and  3y - 4x = -13.\r\n" );
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document.write( "The plots are shown in the Figure below.\r\n" );
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document.write( "    \r\n" );
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document.write( "    Parabola  y =  (red) and tangent lines 3y-4x = -13 (green)  and  3y-10x = -55 (blue)\r\n" );
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\n" ); document.write( "\n" ); document.write( "Notice that I edited the post to make the problem's formulation mathematically clear.\r
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\n" ); document.write( "\n" ); document.write( "Your original formulation was far from to be perfect.\r
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