document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192295>Question 1192295</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The graph of $(x-3)^2 + (y-5)^2=16$ is reflected over the line $y=2$. The new graph is the graph of the equation $x^2 + Bx + y^2 + Dy + F = 0$ for some constants $B$, $D$, and $F$. Find $B+D+F$. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824207 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=824207\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The circle template in general is<BR>\n" );
document.write( "(x-h)^2 + (y-k)^2 = r^2<BR>\n" );
document.write( "with (h,k) as the center and r = radius.\r<BR>\n" );
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document.write( "For this particular problem<BR>\n" );
document.write( "(x-3)^2+(y-5)^2 = 16<BR>\n" );
document.write( "we have (h,k) = (3,5) as the center and r = 4 as the radius.\r<BR>\n" );
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document.write( "The point (3,5) is exactly 3 units above the horizontal line y = 2<BR>\n" );
document.write( "When reflecting over this line, we'll move down 3 units to arrive at the mirror line itself, then move another 3 units further south to get to (3,-1). <BR>\n" );
document.write( "Overall we moved the center down 6 units.<BR>\n" );
document.write( "The radius stays the same because lengths and distances are preserved under reflection operations. \r<BR>\n" );
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document.write( "Therefore we go from <BR>\n" );
document.write( "(x-3)^2+(y-5)^2 = 16<BR>\n" );
document.write( "to<BR>\n" );
document.write( "(x-3)^2+(y+1)^2 = 16<BR>\n" );
document.write( "after reflecting the circle over the horizontal line y = 2.\r<BR>\n" );
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document.write( "Let's expand things out and get everything to one side.<BR>\n" );
document.write( "(x-3)^2+(y+1)^2 = 16<BR>\n" );
document.write( "x^2-6x+9+y^2+2y+1 = 16<BR>\n" );
document.write( "x^2-6x+9+y^2+2y+1-16 = 0<BR>\n" );
document.write( "x^2-6x+y^2+2y-6 = 0\r<BR>\n" );
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document.write( "Comparing that last equation to<BR>\n" );
document.write( "x^2 + Bx + y^2 + Dy + F = 0<BR>\n" );
document.write( "shows that<BR>\n" );
document.write( "B = -6<BR>\n" );
document.write( "D = 2<BR>\n" );
document.write( "F = -6\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "B+D+F = -6+2+(-6) = -10\r<BR>\n" );
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document.write( "Answer: <font color=red>-10</font><BR>\n" );
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