document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=605623>Question 605623</A>: <I><SPAN STYLE=\"word-break: break-all;\"> locate the center, foci, vertices, and ends of the latera recta of the ellipse. find the equation of the ellipse satisfying the given conditions.<BR>\n" );
document.write( "find the equation for the specified hyperbola center at the origin, latus rectum 64/3, eccentricity 5/3. pls graph it<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 #381764 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=381764\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> EQUATION FOR THE HYPERBOLA:<BR>\n" );
document.write( "A hyperbola with the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Fa%5E2-y%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"x%5E2%2Fa%5E2-y%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "has an eccentricity of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=sqrt%281%2Bb%5E2%2Fa%5E2%29 ALIGN=MIDDLE  ALT=\"e=sqrt%281%2Bb%5E2%2Fa%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and a latus rectum of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Al=2b%5E2%2Fa ALIGN=MIDDLE  ALT=\"2%2Al=2b%5E2%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b%5E2%2Fa=64%2F3 ALIGN=MIDDLE  ALT=\"2b%5E2%2Fa=64%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2Fa=32%2F3 ALIGN=MIDDLE  ALT=\"b%5E2%2Fa=32%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We also have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%2Bb%5E2%2Fa%5E2%29=5%2F3 ALIGN=MIDDLE  ALT=\"sqrt%281%2Bb%5E2%2Fa%5E2%29=5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Bb%5E2%2Fa%5E2=25%2F9 ALIGN=MIDDLE  ALT=\"1%2Bb%5E2%2Fa%5E2=25%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2Fa%5E2=25%2F9-1 ALIGN=MIDDLE  ALT=\"b%5E2%2Fa%5E2=25%2F9-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2Fa%5E2=16%2F9 ALIGN=MIDDLE  ALT=\"b%5E2%2Fa%5E2=16%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Fa=4%2F3 ALIGN=MIDDLE  ALT=\"b%2Fa=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Combining both equations:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%5E2%2Fa%29%2F%28b%2Fa%29=%2832%2F3%29%2F%284%2F3%29 ALIGN=MIDDLE  ALT=\"%28b%5E2%2Fa%29%2F%28b%2Fa%29=%2832%2F3%29%2F%284%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=8%29 ALIGN=MIDDLE  ALT=\"highlight%28b=8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2F%28b%2Fa%29=8%2F%284%2F3%29=8%2A3%2F4 ALIGN=MIDDLE  ALT=\"b%2F%28b%2Fa%29=8%2F%284%2F3%29=8%2A3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a=6%29 ALIGN=MIDDLE  ALT=\"highlight%28a=6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That gives us the equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F6%5E2-y%5E2%2F8%5E2=1 ALIGN=MIDDLE  ALT=\"x%5E2%2F6%5E2-y%5E2%2F8%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x%5E2%2F36-y%5E2%2F64=1%29 ALIGN=MIDDLE  ALT=\"highlight%28x%5E2%2F36-y%5E2%2F64=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "GRAPHING:<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=6 ALIGN=MIDDLE  ALT=\"a=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=8 ALIGN=MIDDLE  ALT=\"b=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can draw that box that gives us the vertices and the asymptotes.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-10%2C10%2C-10%2C10%2C%0D%0Agrid%281%29%2C%0D%0Ared%28rectangle%28-6%2C-8%2C6%2C8%29%29%2C%0D%0Ablue%28line%28-9%2C-12%2C9%2C12%29%29%2C%0D%0Ablue%28line%28-9%2C12%2C9%2C-12%29%29%2C%0D%0Ablue%28circle%286%2C0%2C0.4%29%29%2Cblue%28circle%28-6%2C0%2C0.4%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We also know that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=c%2Fa ALIGN=MIDDLE  ALT=\"e=c%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F3=c%2F6 ALIGN=MIDDLE  ALT=\"5%2F3=c%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28c=10%29 ALIGN=MIDDLE  ALT=\"highlight%28c=10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which tells us that the foci are at distance 10 from the center.<BR>\n" );
document.write( "And since the latus rectum is 64/3, there are points of the hyperbola, 32/3 above and below the foci.<BR>\n" );
document.write( "That gives us the location of the foci and 4 more points of the hyperbola<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-25%2C25%2C-25%2C25%2C%0D%0Agrid%281%29%2C%0D%0Ared%28rectangle%28-6%2C-8%2C6%2C8%29%29%2C%0D%0Ablue%28line%28-21%2C-28%2C21%2C28%29%29%2C%0D%0Ablue%28line%28-21%2C28%2C21%2C-28%29%29%2C%0D%0Ablue%28circle%286%2C0%2C1%29%29%2Cblue%28circle%28-6%2C0%2C1%29%29%2C%0D%0Agreen%28circle%2810%2C0%2C1%29%29%2Cgreen%28circle%28-10%2C0%2C1%29%29%2C%0D%0Ablue%28circle%2810%2C32%2F3%2C1%29%29%2Cblue%28circle%28-10%2C32%2F3%2C1%29%29%2C%0D%0Ablue%28circle%2810%2C-32%2F3%2C1%29%29%2Cblue%28circle%28-10%2C-32%2F3%2C1%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > I have the foci (green circles) and six points of the hyperbola (blue circles marking the vertices and the ends of the latera recta).<BR>\n" );
document.write( "Now, I would just connect the points of the hyperbola that I found with smooth curved arches ) ( that hug the asymptotes towards their ends.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-25%2C25%2C-25%2C25%2C4x%2F3%2C-4x%2F3%2Cx%5E2%2F36-y%5E2%2F64%3E1%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-25%2C25%2C-25%2C25%2C4x%2F3%2C-4x%2F3%2Cx%5E2%2F36-y%5E2%2F64%3E1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );