document.write( "Question 1114951: Find the equation of the straight line perpedicular to line y =x^ 2+ 1,and passing through the point (4,5). \n" ); document.write( "

Algebra.Com's Answer #729924 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "The solution by @josgarithmetic is incorrect.\r
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document.write( "First, the given point (4,5) does not belong to the parabola y = x^2 + 1.\r\n" );
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document.write( "     Everybody can check it in 3 seconds.\r\n" );
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document.write( "Therefore, it is INCORRECT to evaluate the slope of 2x at this point.\r\n" );
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document.write( "This slope must be evaluated at the point ON THE CURVE, instead.\r\n" );
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document.write( "So, my solution starts stating that 2x is the slope of the tangent line at the (unknown) point on the parabola y = x^2+1.\r\n" );
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document.write( "Therefore, the slope of the perpendicular line to the curve is .\r\n" );
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document.write( "Now I can write the equation for the perpendicular line having the slope of  and passing through the points (x,y) and (4,5),  \r\n" );
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document.write( "where x is unknown and y = x^2+1:\r\n" );
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document.write( " = ,\r\n" );
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document.write( " = ,\r\n" );
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document.write( " = -x+4\r\n" );
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document.write( " = 0.\r\n" );
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document.write( "The plot of the last polynomial is shown in the Figure 1 below.\r\n" );
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document.write( "Figure 1.  Plot y =  \r\n" );
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document.write( "The polynomial has 3 roots, and the roots are irrational numbers (they are not rational).\r\n" );
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document.write( "I used Excel \"What-if-Analysis\" and calculated these roots approximately, with 3 decimals after the decimal point.\r\n" );
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document.write( "The roots are  x= 2.109,    -0.650  and  -1.46,   giving the values for the slopes  m = \r\n" );
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document.write( "               m = -0.237,  0.769   and  0.343  respectively.\r\n" );
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document.write( "So, the three lines satisfying the condition of perpendicularity are\r\n" );
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document.write( "    y = -0.237*(x-4)+5,   y = 0.769*(x-4)+5,  and   y = 0.343*(x-4)+5.\r\n" );
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document.write( "These lines, together with the parabola  y = x^2+1,  are shown in the Figure 2 below.\r\n" );
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document.write( "Figure 2. Plot y =  (red), y =  -0.237*(x-4)+5 (green), y =  0.769*(x-4)+5 (blue), y =  0.343*(x-4)+5 (purple)\r\n" );
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document.write( "Notice that the intersection of these straight lines is exactly the given point (4,5).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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