document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208761>Question 1208761</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dear Genii<BR>\n" );
document.write( "I have what I fear is a rather silly question.  The other day I came across this question on a GCSE maths paper:<BR>\n" );
document.write( "x-y = k<BR>\n" );
document.write( "x^2+y^2-9=0\r<BR>\n" );
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document.write( "These simultaneous equations have exactly one set of solutions.<BR>\n" );
document.write( "Show that k = +/_3root2 (I don't know hoe to put the square root sign in.\r<BR>\n" );
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document.write( "I have never come across questions like this before and I have no idea what this method/technique is called; I have no idea why it is used and I have no idea about how to solve such questions.  I would very much appreciate it if someone could tell me what this technique is called, why it is used and where I might find instructions about how to use it.\r<BR>\n" );
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document.write( "Marcus Clayson </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847194 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</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=847194\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "sqrt is shorthand for \"square root\"<BR>\n" );
document.write( "example: sqrt(5) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "x-y = k solves to y = x-k\r<BR>\n" );
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document.write( "Plug this into the other equation to see what happens.<BR>\n" );
document.write( "x^2 + y^2 - 9 = 0<BR>\n" );
document.write( "x^2 + (<font color=blue>y</font>)^2 - 9 = 0<BR>\n" );
document.write( "x^2 + (<font color=blue>x-k</font>)^2 - 9 = 0<BR>\n" );
document.write( "x^2 + x^2 - 2kx + k^2 - 9 = 0<BR>\n" );
document.write( "2x^2 - 2kx + k^2 - 9 = 0\r<BR>\n" );
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document.write( "In terms of the variable x we have this quadratic template.<BR>\n" );
document.write( "ax^2 + bx + c = 0<BR>\n" );
document.write( "where in this case,<BR>\n" );
document.write( "a = 2<BR>\n" );
document.write( "b = -2k<BR>\n" );
document.write( "c = k^2 - 9\r<BR>\n" );
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document.write( "If the original system has one solution for a fixed specific value of k, then the equation  2x^2 - 2kx + k^2 - 9 = 0 must have one solution. <BR>\n" );
document.write( "A quadratic having one solution would only happen when the discriminant is equal to 0.<BR>\n" );
document.write( "d = b^2 - 4ac = discriminant\r<BR>\n" );
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document.write( "b^2 - 4ac = 0<BR>\n" );
document.write( "(-2k)^2 - 4(2)(k^2-9) = 0 .......... plug in the a,b,c values mentioned earlier<BR>\n" );
document.write( "4k^2 - 8(k^2-9) = 0<BR>\n" );
document.write( "4k^2 - 8k^2 + 72 = 0<BR>\n" );
document.write( "-4k^2 + 72 = 0<BR>\n" );
document.write( "4k^2 - 72 = 0<BR>\n" );
document.write( "4k^2 = 72<BR>\n" );
document.write( "k^2 = 72/4<BR>\n" );
document.write( "k^2 = 18<BR>\n" );
document.write( "k = plus minus sqrt(18)<BR>\n" );
document.write( "k = sqrt(18) or k = -sqrt(18)<BR>\n" );
document.write( "k = sqrt(9*2) or k = -sqrt(9*2)<BR>\n" );
document.write( "k = sqrt(9)*sqrt(2) or k = -sqrt(9)*sqrt(2)<BR>\n" );
document.write( "<font color=red>k = 3*sqrt(2) or k = -3*sqrt(2)</font>\r<BR>\n" );
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document.write( "Below is a graph of when k = 3*sqrt(2)<BR>\n" );
document.write( "This makes the line x-y = k tangent to the circle toward the bottom right portion of the circle.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-5%2C5%2C-100%2C-100%2Cy+=+x-3%2Asqrt%282%29%29%2C%0D%0Acircle%280%2C0%2C3%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The circle is centered at (0,0) and has radius 3.<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> and <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> are two of many graphing tools I recommend to use.\r<BR>\n" );
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document.write( "And here is the graph when k = -3*sqrt(2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-5%2C5%2C-100%2C-100%2Cy+=+x%2B3%2Asqrt%282%29%29%2C%0D%0Acircle%280%2C0%2C3%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We have another situation where the line is tangent to the circle. <BR>\n" );
document.write( "That tangent point is the solution to the system.\r<BR>\n" );
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document.write( "If  -3*sqrt(2) < k < 3*sqrt(2), then the line x-y = k intersects the circle at two different locations. \r<BR>\n" );
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document.write( "If k < -3*sqrt(2) or k > 3*sqrt(2), then the line doesn't intersect the circle at all and there are no solutions. \r<BR>\n" );
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document.write( "Here is an interactive Desmos graph to try out<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/zijda3qbgr\">https://www.desmos.com/calculator/zijda3qbgr</a><BR>\n" );
document.write( "Move the slider for parameter k so you can see how the blue line moves around. \r<BR>\n" );
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document.write( "3*sqrt(2) = 4.24264 approximately<BR>\n" );
document.write( "</font> </SPAN>\n" );
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