document.write( "Question 1201007: Find the equations for each line that passes through the point (4,-1) and is tangent to the curve y=x^2+4x+3. \n" ); document.write( "
Algebra.Com's Answer #835243 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The equation of a line through a point (x1,y1)\r\n" );
document.write( "with slope m is \"y-y%5B1%5D=m%28x-x%5B1%5D%29\"\r\n" );
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document.write( "The equation of every line through (4,-1) with slope m is\r\n" );
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document.write( "\"y-%28-1%29=m%28x-%284%29%5E%22%22%29\"\r\n" );
document.write( "\"y%2B1=mx-4m\"\r\n" );
document.write( "\"y=mx-4m-1\"\r\n" );
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document.write( "Usually if a line intersects a parabola, it intersects it in TWO points\r\n" );
document.write( "However, a tangent line must intersect a parabola in only ONE (not TWO!)\r\n" );
document.write( "points.  So when we solve the system of the parabola and the line:\r\n" );
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document.write( "\"system%28y=x%5E2%2B4x%2B3%2Cy=mx-4m-1%29\"\r\n" );
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document.write( "we must get only ONE solution.  We set the right sides equal:\r\n" );
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document.write( "\"x%5E2%2B4x%2B3=mx-4m-1\"\r\n" );
document.write( "\"x%5E2%2B4x-mx%2B4%2B4m%2B4=0\"\r\n" );
document.write( "\"x%5E2%2B%284-m%29x%2B%284m%2B4%29=0\"\r\n" );
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document.write( "To guarantee this has only ONE solution, the discriminant must be 0.\r\n" );
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document.write( "\"Discriminant\"\"%22%22=%22%22\"\"b%5E2-4ac\", a=1, b=4-m, c=4m+4.\r\n" );
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document.write( "\"Discriminant\"\"%22%22=%22%22\"\"%284-m%29%5E2-4%2A1%2A%284m%2B4%29\"\"%22%22=%22%22\"\"16-8m%2Bm%5E2-16m-16\"\"%22%22=%22%22\"\"m%5E2-24m\"\r\n" );
document.write( "So we set this discriminant equal to 0:\r\n" );
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document.write( "\"m%5E2-24m=0\"\r\n" );
document.write( "\"m%28m-24%29=0\"\r\n" );
document.write( "m=0; m-24=0\r\n" );
document.write( "       m=24\r\n" );
document.write( "So the equations of the tangent lines are\r\n" );
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document.write( "\"y=mx-4m-1\" with m=0, and m=24\r\n" );
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document.write( "\"y=0x-4%280%29-1\" and \"y=24x-4%2824%29-1\"\r\n" );
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document.write( "\"y=-1\" and \"y=24x-97\"   <---answers\r\n" );
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document.write( "The two green lines are the tangent lines.  The points of tangency are at the vertex of the parabola (-2,-1) and the point (10,143)\r\n" );
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document.write( "Edwin
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