document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=515834>Question 515834</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Complete the square to determine whether the equation<BR>\n" );
document.write( "represents an ellipse, a parabola, a hyperbola, or a degenerate<BR>\n" );
document.write( "conic. If the graph is an ellipse, find the center, foci, vertices,<BR>\n" );
document.write( "and lengths of the major and minor axes. If it is a parabola, find<BR>\n" );
document.write( "the vertex, focus, and directrix. If it is a hyperbola, find the center,<BR>\n" );
document.write( "foci, vertices, and asymptotes. Then sketch the graph of the<BR>\n" );
document.write( "equation. If the equation has no graph, explain why.\r<BR>\n" );
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document.write( "x^2-y^2 = 10(x-y)+1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #344307 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=344307\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It is a hyperbola, find the center,<BR>\n" );
document.write( "foci, vertices, and asymptotes.<BR>\n" );
document.write( "..<BR>\n" );
document.write( "x^2-y^2 = 10(x-y)+1<BR>\n" );
document.write( "x^2-y^2 = 10x-10y+1<BR>\n" );
document.write( "x^2-10x-y^2+10y=1<BR>\n" );
document.write( "complete the square<BR>\n" );
document.write( "(x^2-10x+25)-(y^2-10y+25)=1+25-25=1<BR>\n" );
document.write( "(x-5)^2-(y-5)^2=1<BR>\n" );
document.write( "This is an equation of a hyperbola with horizontal transverse axis of the standard form:<BR>\n" );
document.write( "(x-h)^2/a^2-(y-k)^2/b^2=1, with (h,k) being the (x,y) coordinates of the center:<BR>\n" );
document.write( "For given equation:<BR>\n" );
document.write( "center: (5, 5)<BR>\n" );
document.write( "a^2=1<BR>\n" );
document.write( "a=1<BR>\n" );
document.write( "b^2=1<BR>\n" );
document.write( "b=1<BR>\n" );
document.write( "c^2=a^2+b^2=1+1=2<BR>\n" );
document.write( "c=&#8730;2<BR>\n" );
document.write( "..<BR>\n" );
document.write( "foci: (5±c,5)=(5±&#8730;2,5)=(5+&#8730;2,5) and (5-&#8730;2,5)<BR>\n" );
document.write( "vertices: (5±a,5)=(5±1,5)=(6,5) and (4,5)<BR>\n" );
document.write( "slope:± b/a=±1/1=±1<BR>\n" );
document.write( "equation of asymptotes: y=mx+b=±(1)x+b<BR>\n" );
document.write( "equation for slope=-1<BR>\n" );
document.write( "y=-x+b<BR>\n" );
document.write( "solving for b using (x,y) coordinates of center thru which asymptotes pass<BR>\n" );
document.write( "5=-5+b<BR>\n" );
document.write( "b=10<BR>\n" );
document.write( "equation: y=-x+10<BR>\n" );
document.write( "..<BR>\n" );
document.write( "equation for slope=1<BR>\n" );
document.write( "y=x+b<BR>\n" );
document.write( "solving for b using (x,y) coordinates of center thru which asymptotes pass<BR>\n" );
document.write( "5=5+b<BR>\n" );
document.write( "b=<BR>\n" );
document.write( "equation: y=x<BR>\n" );
document.write( "..<BR>\n" );
document.write( "see graph below as a visual check on the answers above:<BR>\n" );
document.write( "y=±((x-5)^2-1)^.5+5\r<BR>\n" );
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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%28%28x-5%29%5E2-1%29%5E.5%2B5%2C-%28%28x-5%29%5E2-1%29%5E.5%2B5%2C-x%2B10%2Cx%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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