document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197351>Question 1197351</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation (hyperbola) if the<BR>\n" );
document.write( "asymptotes y  -  3 = √13/6 (x-2) and focus at (9, 3) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830604 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=830604\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "If you use the \"√\" symbol for a square root, then use parentheses to make it clear exactly what part of the expression is under the radical.  \"√13/6\" could mean either sqrt(13)/6 or sqrt(13/6).<br><BR>\n" );
document.write( "A coefficient like sqrt(13/6) would probably not be given in a problem, since it is not in simplified form; so I will assume the coefficient is sqrt(13)/6.<br><BR>\n" );
document.write( "You show the equation of one of the asymptotes; I further assume that the asymptotes (plural) have equations with coefficients sqrt(13)/6 and -sqrt(13)/6.<br><BR>\n" );
document.write( "The equations of the asymptotes tell us the center of the hyperbola is (2,3); and the focus at (9,3) tells us the branches of the hyperbola open right and left.  So we know the equation of the hyperbola has the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2%2Fa%5E2-%28y-3%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2%2Fa%5E2-%28y-3%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The slopes of the asymptotes tell us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Fa=sqrt%2813%29%2F6 ALIGN=MIDDLE  ALT=\"b%2Fa=sqrt%2813%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Finally, with the center at (2,3) and one focus at (9,3), we know that c, the distance from the center to each focus, is 7.<br><BR>\n" );
document.write( "So we have...<BR>\n" );
document.write( "b/a = sqrt(13)/6;<BR>\n" );
document.write( "c = 7;<BR>\n" );
document.write( "and, for a hyperbola, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2+=+a%5E2%2Bb%5E2 ALIGN=MIDDLE  ALT=\"c%5E2+=+a%5E2%2Bb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then the simple observation that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5E2+=+sqrt%2813%29%5E2%2B6%5E2 ALIGN=MIDDLE  ALT=\"7%5E2+=+sqrt%2813%29%5E2%2B6%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > tells us that b = sqrt(13) and a = 6.<br><BR>\n" );
document.write( "So the equation of the hyperbola is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2%2F36-%28y-3%29%5E2%2F13=1 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2%2F36-%28y-3%29%5E2%2F13=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );