document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1150043>Question 1150043</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Write an equation for the hyperbola that has eccentricity 2, center at (0,0), and vertex at (0,8) <BR>\n" );
document.write( "An equation for the hyperbola is...? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854697 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5398)</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=854697\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> FACTS ABOUT HYPERBOLAS:<BR>\n" );
document.write( "The equation for a hyperbola centered at (h,k) can be written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fp%5E2-%28y-k%29%5E2%2Fq%5E2=%22+%22%2B-+1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fp%5E2-%28y-k%29%5E2%2Fq%5E2=%22+%22%2B-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some pair (p,q) of positive numbers.<BR>\n" );
document.write( "If a given equation can be written as above, you would have the values of h, k, p, and q and the sign for the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22+%22%2B-+1 ALIGN=MIDDLE  ALT=\"%22+%22%2B-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > part.<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fp%5E2-%28y-k%29%5E2%2Fq%5E2=-1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fp%5E2-%28y-k%29%5E2%2Fq%5E2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-k%29%5E2%2Fq%5E2-%28x-h%29%5E2%2Fp%5E2=1 ALIGN=MIDDLE  ALT=\"%28y-k%29%5E2%2Fq%5E2-%28x-h%29%5E2%2Fp%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-k%29%5E2%2Fq%5E2=%28x-h%29%5E2%2Fp%5E2%2B1 ALIGN=MIDDLE  ALT=\"%28y-k%29%5E2%2Fq%5E2=%28x-h%29%5E2%2Fp%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , you see that it must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-k%29%5E2%2Fq%5E2%3E=1 ALIGN=MIDDLE  ALT=\"%28y-k%29%5E2%2Fq%5E2%3E=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28y-k%29%3E=q ALIGN=MIDDLE  ALT=\"abs%28y-k%29%3E=q\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y-k%3E=q%2C%22or%22%2Cy-k%3C=-q%29 ALIGN=MIDDLE  ALT=\"system%28y-k%3E=q%2C%22or%22%2Cy-k%3C=-q%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y%3E=k%2Bq%2C%22or%22%2Cy%3C=k-q%29 ALIGN=MIDDLE  ALT=\"system%28y%3E=k%2Bq%2C%22or%22%2Cy%3C=k-q%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the graph will have an upper branch and a lower branch, like this: <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C1%2C19%2C1%2C19%2Ccircle%2810%2C15%2C0.1%29%2Ccircle%2810%2C5%2C0.1%29%2C%0D%0Alocate%289.8%2C16%2CF%29%2Clocate%289.8%2C4.7%2C%22F%27%22%29%2Clocate%2812%2C9.95%2Cp%29%2C%0D%0Alocate%2810.2%2C12%2Cq%29%2Clocate%2810.2%2C9%2Cq%29%2Clocate%2812%2C11.7%2Cc%29%2C%0D%0Arectangle%286%2C7%2C10%2C13%29%2Crectangle%286%2C7%2C14%2C10%29%2Crectangle%286%2C7%2C14%2C13%29%2C%0D%0Aarc%2810%2C10%2C10%2C10%2C-150%2C-32%29%2Carc%2810%2C10%2C10%2C10%2C32%2C150%29%2C%0D%0Agraph%28300%2C300%2C1%2C19%2C1%2C19%2C2.5%2B0.75x%2C17.5-0.75x%2C10%2Bsqrt%289%2B0.5625%28x-10%29%5E2%29%2C%0D%0A10-sqrt%289%2B0.5625%28x-10%29%5E2%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "On the other hand, for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fp%5E2-%28y-k%29%5E2%2Fq%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fp%5E2-%28y-k%29%5E2%2Fq%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fp%5E2=%28y-k%29%5E2%2Fq%5E2%2B1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fp%5E2=%28y-k%29%5E2%2Fq%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so it must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fp%5E2%3E=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fp%5E2%3E=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x-h%29%3E=p ALIGN=MIDDLE  ALT=\"abs%28x-h%29%3E=p\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x-h%3E=p%2C%22or%22%2Cx-h%3C=-p%29 ALIGN=MIDDLE  ALT=\"system%28x-h%3E=p%2C%22or%22%2Cx-h%3C=-p%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%3E=h%2Bp%2C%22or%22%2Cy%3C=h-p%29 ALIGN=MIDDLE  ALT=\"system%28x%3E=h%2Bp%2C%22or%22%2Cy%3C=h-p%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so there is a left branch and a right branch to the graph, like this:<BR>\n" );
document.write( ": <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C1%2C19%2C1%2C19%2C%0D%0Arectangle%286%2C7%2C10%2C13%29%2Crectangle%286%2C7%2C14%2C10%29%2Crectangle%286%2C7%2C14%2C13%29%2C%0D%0Agraph%28300%2C300%2C1%2C19%2C1%2C19%2C2.5%2B0.75x%2C17.5-0.75x%2C10%2Bsqrt%280.5625%28x-10%29%5E2-9%29%2C%0D%0A10-sqrt%280.5625%28x-10%29%5E2-9%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In either case, the red and green lines are the asymptotes, with slopes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%2Fp ALIGN=MIDDLE  ALT=\"q%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-q%2Fp ALIGN=MIDDLE  ALT=\"-q%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Textbooks would write the equation of a hyperbola as<BR>\n" );
document.write( "either <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;\" > for a hyperbola with upper and lower branches with vertices at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=k+%2B-+a ALIGN=MIDDLE  ALT=\"y=k+%2B-+a\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a hyperbola with upper and lower branches with vertices at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=h+%2B-+a ALIGN=MIDDLE  ALT=\"x=h+%2B-+a\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "When you write the equation you have number values instead of letters,<BR>\n" );
document.write( "but your teacher may expect you to use the same letters they use.<BR>\n" );
document.write( " <BR>\n" );
document.write( "COMPARING THOSE FACTS WITH YOUR QUESTION:<BR>\n" );
document.write( "The hyperbola in your question has its center at (0,0), and vertex at (0,8), directly above its center by 8 units.<BR>\n" );
document.write( "It is just like the hyperbola with equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-k%29%5E2%2Fq%5E2-%28x-h%29%5E2%2Fp%5E2=1 ALIGN=MIDDLE  ALT=\"%28y-k%29%5E2%2Fq%5E2-%28x-h%29%5E2%2Fp%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <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;\" > shown in the top drawing above.<BR>\n" );
document.write( "The center for the hyperbola in the equation above was (h,k), so for the hyperbola in your question, (h,k) is (0,0), with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=k=0 ALIGN=MIDDLE  ALT=\"h=k=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The distance between the center and the vertex above was labeled as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q ALIGN=MIDDLE  ALT=\"q\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the drawing, so for the hyperbola in your question, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q=8 ALIGN=MIDDLE  ALT=\"q=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=8 ALIGN=MIDDLE  ALT=\"a=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "All you need to write your equation is the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "ABOUT ECCENTRICITY:<BR>\n" );
document.write( "So, what do we know about eccentricity?<BR>\n" );
document.write( "It is defined for ellipses and hyperbolas as a ratio of distances from the center, distance to a focus divided by distance to a vertex to the focus.<BR>\n" );
document.write( "and for both curves the drawings have right triangles with sides labeled <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q ALIGN=MIDDLE  ALT=\"q\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the hypotenuse and the distance from center to focus.<BR>\n" );
document.write( "According to the Pythagorean theorem, for those triangles <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2=p%5E2%2Bq%5E2 ALIGN=MIDDLE  ALT=\"c%5E2=p%5E2%2Bq%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <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>\n" );
document.write( "In the drawing above, the triangle side labeled c is not the line segment from the center to a focus, but I draw an arc of circle to show that the lengths are the same.<BR>\n" );
document.write( " <BR>\n" );
document.write( "FINDING <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > AND <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "With the equation written as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2F8%5E2-x%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"y%5E2%2F8%5E2-x%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=with+a=8 ALIGN=MIDDLE  ALT=\"with+a=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the eccentricity would be <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=2=c%2F8 ALIGN=MIDDLE  ALT=\"2=c%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A8=c ALIGN=MIDDLE  ALT=\"2%2A8=c\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28c=16%29 ALIGN=MIDDLE  ALT=\"highlight%28c=16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting the values found got <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<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;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%5E2=8%5E2%2Bb%5E2 ALIGN=MIDDLE  ALT=\"16%5E2=8%5E2%2Bb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=16%5E2-8%5E2=256-64=highlight%2892%29 ALIGN=MIDDLE  ALT=\"b%5E2=16%5E2-8%5E2=256-64=highlight%2892%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=sqrt%28192%29=8sqrt%283%29 ALIGN=MIDDLE  ALT=\"b=sqrt%28192%29=8sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >and the equation for the hyperbola in the question,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2F8%5E2-x%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"y%5E2%2F8%5E2-x%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > turns into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y%5E2%2F64-x%5E2%2F192=1%29 ALIGN=MIDDLE  ALT=\"highlight%28y%5E2%2F64-x%5E2%2F192=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The graph is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-20%2C20%2C-20%2C20%2C%0D%0Arectangle%28-13.86%2C-8%2C13.86%2C8%29%2Clocate%285%2C8%2Cb%29%2C%0D%0Alocate%280.6%2C10%2CV%29%2Clocate%280.6%2C-8.2%2C%22V%27%22%29%2C%0D%0Alocate%280.6%2C14.5%2CF%29%2Clocate%280.6%2C1-13.5%2C%22F%27%22%29%2C%0D%0Alocate%2813.9%2C4.5%2Ca%29%2Clocate%2813.9%2C-3%2Ca%29%2Clocate%287%2C4.5%2Cc%29%2C%0D%0Agraph%28300%2C300%2C-20%2C20%2C-20%2C20%2C0.5774x%2C-0.5774x%2C8sqrt%281%2Bx%5E2%2F192%29%2C%0D%0A-8sqrt%281%2Bx%5E2%2F192%29%29%2C%0D%0Acircle%280%2C8%2C0.5%29%2Ccircle%280%2C-8%2C0.5%29%2C%0D%0Acircle%280%2C13.86%2C0.5%29%2Ccircle%280%2C-13.86%2C0.5%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > . The asymptotes' slopes are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28a%2Fb=8%2F8sqrt%283%29=1%2Fsqrt%283%29=sqrt%283%29%2F3%29 ALIGN=MIDDLE  ALT=\"red%28a%2Fb=8%2F8sqrt%283%29=1%2Fsqrt%283%29=sqrt%283%29%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28-a%2Fb=-1%2Fsqrt%283%29=-sqrt%283%29%2F3%29 ALIGN=MIDDLE  ALT=\"green%28-a%2Fb=-1%2Fsqrt%283%29=-sqrt%283%29%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );