document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209658>Question 1209658</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The real numbers x and y satisfy<BR>\n" );
document.write( "x^2 + y^2 - 8x + 6y + 23 = 0.<BR>\n" );
document.write( "Find the largest possible value of x + y. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849794 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=849794\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>3</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Explanation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you complete the square for the x and y terms, then you'll go from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+y%5E2+-+8x+%2B+6y+%2B+23+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+y%5E2+-+8x+%2B+6y+%2B+23+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%5E2%2B%28y%2B3%29%5E2+=+2 ALIGN=MIDDLE  ALT=\"%28x-4%29%5E2%2B%28y%2B3%29%5E2+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is a circle with center (4,-3). The radius is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+sqrt%282%29 ALIGN=MIDDLE  ALT=\"r+=+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can use various tools such as GeoGebra or WolframAlpha to verify the claim. <BR>\n" );
document.write( "Or you can expand out the terms in the 2nd equation to arrive back at the 1st equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This will mean the point (x,y) is on the circle's boundary.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Consider the equation x+y = k<BR>\n" );
document.write( "The goal is to find the largest k value possible.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you draw out the graph of the circle, and draw various lines of the form x+y = k, then you should notice that exactly one line will be tangent to the circle at the northeast corner.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-2%2C6%2C-6%2B1.5%2C2%2B1.5%2C%0D%0Agraph%28400%2C400%2C-2%2C6%2C-6%2B1.5%2C2%2B1.5%2C-100%2C-x%2B3%29%2C%0D%0Acircle%284%2C-3%2Csqrt%282%29%29%2C%0D%0Acircle%284%2C-3%2C0.05%29%2Ccircle%284%2C-3%2C0.07%29%2C%0D%0Acircle%285%2C-2%2C0.05%29%2Ccircle%285%2C-2%2C0.07%29%2C%0D%0Alocate%284%2B0.1%2C-3-0.1%2C%22A%22%29%2Clocate%285%2B0.1%2C-2%2B0.2%2C%22B%22%29%2Clocate%281.5%2C2%2C%22x%2By=3%22%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A = center of the circle = (4,-3)<BR>\n" );
document.write( "B = tangent point = (5,-2)<BR>\n" );
document.write( "The green tangent line has the equation x+y = <font color=red>3</font><BR>\n" );
document.write( "This is the largest x+y can get when subjected to the condition that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+y%5E2+-+8x+%2B+6y+%2B+23+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+y%5E2+-+8x+%2B+6y+%2B+23+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > aka <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%5E2%2B%28y%2B3%29%5E2+=+2 ALIGN=MIDDLE  ALT=\"%28x-4%29%5E2%2B%28y%2B3%29%5E2+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Note radius AB has slope 1 which is the negative reciprocal of the tangent slope -1.<BR>\n" );
document.write( "Radius AB is perpendicular to the green tangent line.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check out this interactive <a  rel=nofollow HREF=\"https://www.desmos.com/calculator/ke5vswmyda\">Desmos graph</a><BR>\n" );
document.write( "Adjusting the slider for parameter k will move the line x+y = k up or down.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Verification using <a  rel=nofollow HREF=\"https://www.wolframalpha.com/input?i=max+x%2By+when+x%5E2+%2B+y%5E2+-+8x+%2B+6y+%2B+23+%3D+0\">WolframAlpha</a><BR>\n" );
document.write( "The answer on the WolframAlpha page is a bit buried in a sea of numbers & symbols, but it does mention <font color=red>3</font> under the \"global maximum\" subsection and just before the \"at (x,y) = (5,-2)\"\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "More practice with a similar problem is found <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Quadratic-relations-and-conic-sections.faq.question.1208800.html\">here</a><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );