document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=526311>Question 526311</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the coordinate plane, the graphs of the equations x^2 + y^2 – 4x + 6y –12 =0 and y =  ax^2 + bx + c  have exactly  3 points  in common . Two of these points are (3, -3) and (7, -3). What are all possible coordinates of the third point?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #348420 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=348420\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Nope.  (3,-3) is NOT on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\ +\ y^2\ –\ 4x\ +\ 6y\ –\ 12\ =\ 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (3)^2\ +\ (-3)^2\ –\ 4(3)\ +\ 6(-3)\ –\ 12\ =\ -24\ \neq\ 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "On the other hand (-3,-3) IS on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\ +\ y^2\ –\ 4x\ +\ 6y\ –\ 12\ =\ 0\">, so on the off chance you made the obvious typo:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First complete the square on each of the variables to obtain the standard form of the equation of a circle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\ -\ 2)^\ +\ (y\ +\ 3)^2\ =\ 5^2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(Verification left as an exercise for the student)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note that (-3,-3) and (7,-3) are the endpoints of a horizontal diameter.  Hence, if a parabola, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ ax^2\ +\ bx\ +\ c\"> which has a vertical line axis of symmetry passes through those two diameter endpoints, the third point of commonality, given exactly three points in common, must be the vertex of the parabola which could intersect the circle at either endpoint of a vertical diameter segment.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Find both points on the circle where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 2\"> by evaluating:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ \pm\sqrt{25\ -\ (0)^2}\ -\ 3\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<div  DISABLED_style= \"text-align:center\"><a  rel=nofollow HREF=\"http://outcampaign.org/\" target=\"_blank\"><IMG STYLE=\"max-width:80%;\" src=\"http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png\" border=\"0\" alt=\"The Out Campaign: Scarlet Letter of Atheism\" width=\"143\" height=\"122\" /></a></div><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );