document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1035355>Question 1035355</A>: <I><SPAN STYLE=\"word-break: break-all;\"> write the equation.<BR>\n" );
document.write( "1. hyperbola with foci (0,4) and (0,-4) and asymptotes at y=2x<BR>\n" );
document.write( "2. circle passing through the points (12,1) and (2,-3) with center on the line 2x-5y+10=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #650077 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=650077\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1.The major axis is on the line connecting the foci, which is the y-axis.<BR>\n" );
document.write( "The center of the hyperbola is halfway between the foci, at (0,0), the origin,<BR>\n" );
document.write( "so the minor axis (perpendicular to the y-axis through the center) is the x-axis.<BR>\n" );
document.write( "The equation for a parabola centered at the origin, with the y-axis as its major axis is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2Fa%5E2-x%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"y%5E2%2Fa%5E2-x%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3E0 ALIGN=MIDDLE  ALT=\"a%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%3E0 ALIGN=MIDDLE  ALT=\"b%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The focal distance, <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 half the distance between the foci.<BR>\n" );
document.write( "In this case, it is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=%284-%28-4%29%29%2F2=4 ALIGN=MIDDLE  ALT=\"c=%284-%28-4%29%29%2F2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The semimajor axis, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , is the distance between the center and each vertex.<BR>\n" );
document.write( "The major axis is the vertical segment connecting vertices (0,-a), (0,a),<BR>\n" );
document.write( "which are on the major axis, between the foci, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Ca%3C4 ALIGN=MIDDLE  ALT=\"0%3Ca%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The semi-minor axis is <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( "The asymptotes cross at the center of the hyperbola (in this case (0,0), the origin),<BR>\n" );
document.write( "and have equations <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28a%2Fb%29x ALIGN=MIDDLE  ALT=\"y=%28a%2Fb%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28a%2Fb%29x ALIGN=MIDDLE  ALT=\"-%28a%2Fb%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So far, we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28200%2C400%2C-3%2C3%2C-6%2C6%2Cgrid%280%29%2C%0D%0Atriangle%28-5%2C-10%2C5%2C-10%2C0%2C0%29%2Ctriangle%285%2C10%2C-5%2C10%2C0%2C0%29%2C%0D%0Agreen%28rectangle%28-1.79%2C-3.58%2C1.79%2C3.58%29%29%2C%0D%0Acircle%280%2C-4%2C0.1%29%2Ccircle%280%2C4%2C0.1%29%2C%0D%0Acircle%280%2C-3.58%2C0.1%29%2Ccircle%280%2C3.58%2C0.1%29%2C%0D%0Alocate%28-1.8%2C2%2Cgreen%28a%29%29%2Clocate%28-1.8%2C-1.6%2Cgreen%28a%29%29%2C%0D%0Alocate%281.5%2C2%2Cgreen%28a%29%29%2Clocate%281.5%2C-1.6%2Cgreen%28a%29%29%2C%0D%0Alocate%28-1%2C3.58%2Cgreen%28b%29%29%2Clocate%280.8%2C3.58%2Cgreen%28b%29%29%2C%0D%0Alocate%280.8%2C1.6%2Cc%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The values <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=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > determine a rectangle,<BR>\n" );
document.write( "passing through the vertices,<BR>\n" );
document.write( "with sides measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a ALIGN=MIDDLE  ALT=\"2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b ALIGN=MIDDLE  ALT=\"2b\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and diagonals measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2c ALIGN=MIDDLE  ALT=\"2c\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The axes and the asymptotes divide that rectangle into 8 right trianlges with legs <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=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and hypotenuse <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<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;\" > .<BR>\n" );
document.write( "Since the asymptotes have equations <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x ALIGN=MIDDLE  ALT=\"y=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x ALIGN=MIDDLE  ALT=\"y=-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "in this case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fb=2 ALIGN=MIDDLE  ALT=\"a%2Fb=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2b ALIGN=MIDDLE  ALT=\"a=2b\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=4b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2=4b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=4 ALIGN=MIDDLE  ALT=\"c=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E2=4b%5E2%2Bb%5E2 ALIGN=MIDDLE  ALT=\"4%5E2=4b%5E2%2Bb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16=5b%5E2 ALIGN=MIDDLE  ALT=\"16=5b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=16%2F5 ALIGN=MIDDLE  ALT=\"b%5E2=16%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=4%2A16%2F5=64%2F5 ALIGN=MIDDLE  ALT=\"a%5E2=4%2A16%2F5=64%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the equation for the hyperbola in this problem is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2F%2864%2F5%29-x%5E2%2F%2816%2F5%29=1 ALIGN=MIDDLE  ALT=\"y%5E2%2F%2864%2F5%29-x%5E2%2F%2816%2F5%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2F12.8-x%5E2%2F3.2=1 ALIGN=MIDDLE  ALT=\"y%5E2%2F12.8-x%5E2%2F3.2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y%5E2%2F64-5x%5E2%2F16=1 ALIGN=MIDDLE  ALT=\"5y%5E2%2F64-5x%5E2%2F16=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2F64-c%5E2%2F16=1%2F5 ALIGN=MIDDLE  ALT=\"y%5E2%2F64-c%5E2%2F16=1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );