document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199827>Question 1199827</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi;\r<BR>\n" );
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document.write( "I need help with this multiple choice question.<BR>\n" );
document.write( "For the line y=x to be tangent to the curve with equation y =   k / x - 1<BR>\n" );
document.write( "                                                              (k divided by x-1)\r<BR>\n" );
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document.write( "A. k=1<BR>\n" );
document.write( "B. k>1<BR>\n" );
document.write( "C. k>-1/4<BR>\n" );
document.write( "D. k=1/4<BR>\n" );
document.write( "E. 4k+1<0\r<BR>\n" );
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document.write( "Thanks. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833799 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=833799\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'm assuming all of the \"x-1\" is in the denominator, and that you intend to say <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+k%2F%28x-1%29 ALIGN=MIDDLE  ALT=\"y+=+k%2F%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If so, then you should wrap \"x-1\" in parenthesis and write y = k/(x-1)\r<BR>\n" );
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document.write( "I'll be using differential calculus to solve this problem. <BR>\n" );
document.write( "If you aren't familiar with the topic, then please let me know.\r<BR>\n" );
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document.write( "Let's apply the derivative with respect to x.<BR>\n" );
document.write( "Use the power rule where y = x^n leads to dy/dx = n*x^(n-1)\r<BR>\n" );
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document.write( "In this specific case:<BR>\n" );
document.write( "y = k/(x - 1)<BR>\n" );
document.write( "y = k(x - 1)^(-1)<BR>\n" );
document.write( "dy/dx = -k(x - 1)^(-2)\r<BR>\n" );
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document.write( "The slope of the tangent is found through the derivative dy/dx<BR>\n" );
document.write( "The tangent y = x has a slope of 1, which means dy/dx = 1<BR>\n" );
document.write( "Use this fact to determine k in terms of x. \r<BR>\n" );
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document.write( "dy/dx = -k(x - 1)^(-2)<BR>\n" );
document.write( "1 = -k(x - 1)^(-2)<BR>\n" );
document.write( "(x - 1)^2 = -k<BR>\n" );
document.write( "k = -(x-1)^2<BR>\n" );
document.write( "This will be useful later.<BR>\n" );
document.write( "Let's call this equation (1)\r<BR>\n" );
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document.write( "For the line y = x to be tangent to y = k/(x-1), the two must intersect.\r<BR>\n" );
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document.write( "Apply substitution.<BR>\n" );
document.write( "y = k/(x - 1)<BR>\n" );
document.write( "x = k/(x - 1)<BR>\n" );
document.write( "x(x-1) = k<BR>\n" );
document.write( "x(x-1) = -(x-1)^2 ........ refer to equation (1)<BR>\n" );
document.write( "x^2-x = -(x^2-2x+1)<BR>\n" );
document.write( "x^2-x = -x^2+2x-1<BR>\n" );
document.write( "x^2-x+x^2-2x+1 = 0<BR>\n" );
document.write( "2x^2-3x+1 = 0\r<BR>\n" );
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document.write( "Use the quadratic formula to solve for x.<BR>\n" );
document.write( "I'll skip this subsection of steps.<BR>\n" );
document.write( "You should get x = 1 and x = 1/2 = 0.5\r<BR>\n" );
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document.write( "If x = 1, then,<BR>\n" );
document.write( "k = -(x-1)^2<BR>\n" );
document.write( "k = -(1-1)^2<BR>\n" );
document.write( "k = 0\r<BR>\n" );
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document.write( "If x = 1/2, then,<BR>\n" );
document.write( "k = -(x-1)^2<BR>\n" );
document.write( "k = -(1/2-1)^2<BR>\n" );
document.write( "k = -1/4\r<BR>\n" );
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document.write( "We have either k = 0 or k = -1/4\r<BR>\n" );
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document.write( "If k = 0 happens, then <BR>\n" );
document.write( "y = k/(x - 1)<BR>\n" );
document.write( "becomes<BR>\n" );
document.write( "y = 0/(x - 1)<BR>\n" );
document.write( "y = 0<BR>\n" );
document.write( "this line is completely flat and horizontal. <BR>\n" );
document.write( "It overlaps the x axis.<BR>\n" );
document.write( "Clearly y = x is not tangent to y = 0.<BR>\n" );
document.write( "Therefore, we must rule out k = 0.\r<BR>\n" );
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document.write( "On the other hand if k = -1/4 = -0.25, then,<BR>\n" );
document.write( "y = k/(x - 1)<BR>\n" );
document.write( "y = -0.25/(x - 1)<BR>\n" );
document.write( "y = (-1/4)/(x-1)<BR>\n" );
document.write( "y = -1/(4(x - 1))<BR>\n" );
document.write( "y = -1/(4x - 4)\r<BR>\n" );
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document.write( "I'll leave it to the student to verify the claim that y = x is tangent to y = -1/(4x - 4)<BR>\n" );
document.write( "As another homework exercise, the student should try to find the point of intersection.\r<BR>\n" );
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document.write( "Graph:<BR>\n" );
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document.write( "Graphing tools such as Desmos and GeoGebra are very useful as quick visual verification methods.\r<BR>\n" );
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document.write( "Answer: <font size=4 color=red>k = -1/4</font><BR>\n" );
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