document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=765498>Question 765498</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the vertices, asymptotes, and foci of the hyperbola.\r<BR>\n" );
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document.write( "y^2-9x^2-10y-54x-65=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #466225 by </B><A HREF=/tutors/aboutme.mpl?userid=lwsshak3><B>lwsshak3(11628)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=lwsshak3><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=466225\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the vertices, asymptotes, and foci of the hyperbola. <BR>\n" );
document.write( "y^2-9x^2-10y-54x-65=0<BR>\n" );
document.write( "***<BR>\n" );
document.write( "complete the square:<BR>\n" );
document.write( "y^2-10y-9x^2-54x-65=0<BR>\n" );
document.write( "(y^2-10y+25)-9(x^2+6x+9)=65+25-81<BR>\n" );
document.write( "(y-5)^2-9(x+3)^2=9<BR>\n" );
document.write( "(y-5)^2/9-(x+3)^2=1<BR>\n" );
document.write( "This is an equation of a hyperbola with vertical transverse axis.<BR>\n" );
document.write( "Its standard form: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-k%29%5E2%2Fa%5E2-%28x-h%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28y-k%29%5E2%2Fa%5E2-%28x-h%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, (h,k)=(x,y) coordinates of the center.<BR>\n" );
document.write( "For given hyperbola:<BR>\n" );
document.write( "center: (-3,5)<BR>\n" );
document.write( "a^2=9<BR>\n" );
document.write( "a=&#8730;9=3<BR>\n" );
document.write( "vertices: (-3,5±a)=(-3,5±3)=(-3,2) and (-3,8)<BR>\n" );
document.write( "..<BR>\n" );
document.write( "b^2=1<BR>\n" );
document.write( "b=1<BR>\n" );
document.write( "..<BR>\n" );
document.write( "c^2=a^2+b^2=10<BR>\n" );
document.write( "c&#8776;&#8730;10&#8776;3.16<BR>\n" );
document.write( "foci:  (-3,5±c)=(-3,5±3.16)=(-3,1.84) and (-3,8.16)<BR>\n" );
document.write( "..<BR>\n" );
document.write( "slopes of asymptotes with vertical transverse=±a/b=±3/1=±3<BR>\n" );
document.write( "asymptotes are straight line equations that go thru center of hyperbola<BR>\n" );
document.write( "For asymptote with positive slope:<BR>\n" );
document.write( "y=3x+b<BR>\n" );
document.write( "solve for b using coordinates of center<BR>\n" );
document.write( "5=3*-3+b<BR>\n" );
document.write( "b=14<BR>\n" );
document.write( "equation: y=3x+14<BR>\n" );
document.write( "..<BR>\n" );
document.write( "For asymptote with negative slope:<BR>\n" );
document.write( "y=-3x+b<BR>\n" );
document.write( "solve for b using coordinates of center<BR>\n" );
document.write( "5=-3*-3+b<BR>\n" );
document.write( "b=-4<BR>\n" );
document.write( "equation: y=-3x-4<BR>\n" );
document.write( " </SPAN>\n" );
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