document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=416117>Question 416117</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I do not understand how to find the center, transverse asxis, vertices, foci and asymptotes for the following equations:<BR>\n" );
document.write( "y^2-x^2-4y+4x-1=0 and y^2-4x^2-4y-8x-4=0\r<BR>\n" );
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document.write( "(I think I am supposed to complete the square but I do not know how to do that with two variables) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #291626 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=291626\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I do not understand how to find the center, transverse asxis, vertices, foci and asymptotes for the following equations:<BR>\n" );
document.write( "y^2-x^2-4y+4x-1=0 and y^2-4x^2-4y-8x-4=0\r<BR>\n" );
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document.write( "..<BR>\n" );
document.write( "y^2-x^2-4y+4x-1=0<BR>\n" );
document.write( "completing the square,<BR>\n" );
document.write( "(y^2-4y+4)-(x^2-4x+4)=1+4+4<BR>\n" );
document.write( "(y-2)^2-(x-2)^2=9<BR>\n" );
document.write( "(y-2)^2/9-(x-2)^2/9=1<BR>\n" );
document.write( "Standard form for this hyperbola: (y-k)^2/a^2-(x-h)^2/b^2+1, with (h,k) being the (x,y) coordinates of the center. Since the y-term is first, the hyperbola opens upwards (transverse axis vertical) If the x-term is first, the hyperbola opens sideways and the transverse axis is horizontal.<BR>\n" );
document.write( "..<BR>\n" );
document.write( "center, (2,2)<BR>\n" );
document.write( "transverse axis, x=2<BR>\n" );
document.write( "a^2=9<BR>\n" );
document.write( "a=3 (length of vertices)<BR>\n" );
document.write( "b^2=9<BR>\n" );
document.write( "b=3<BR>\n" );
document.write( "c^=a^2+b^2<BR>\n" );
document.write( "c=sqrt(a^2+b^2)=sqrt(18)=4.34(length of foci)<BR>\n" );
document.write( "foci:(2,6.34),(2,-2.34)<BR>\n" );
document.write( "vertices:(2,5),(2-1)<BR>\n" );
document.write( "Asymptotes: slope of asymptotes = a/b and -a/b= 3/3 and -3/3=1 and -1<BR>\n" );
document.write( "finding equation of asymptotes:<BR>\n" );
document.write( "y=mx+b and y=-mx+b (b is y-intercept)<BR>\n" );
document.write( "using points of center,<BR>\n" );
document.write( "2=2+b<BR>\n" );
document.write( "b=0<BR>\n" );
document.write( "2=-2+b<BR>\n" );
document.write( "b=4<BR>\n" );
document.write( "Equations of asymptotes:<BR>\n" );
document.write( "y=x<BR>\n" );
document.write( "y=-x+4\r<BR>\n" );
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document.write( "The graph below should help you understand the algebra above.  <BR>\n" );
document.write( "..<BR>\n" );
document.write( "y=2+((9+(x-2)^2)^.5<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C+2%2B%28%289%2B%28x-2%29%5E2%29%5E.5%29%2C2-%28%289%2B%28x-2%29%5E2%29%5E.5%29%2Cx%2C-x%2B4%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I'm not going to do the second equation as it pretty much would be a repeat of what I just did. If you get stuck on it send me an email and I will try to help.\r<BR>\n" );
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