document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=874663>Question 874663</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need to find the equation of a hyperbola with foci at (-3,7) and (-3,-5) whose transverse axis is 8 units long. I have no idea where to even start with this, but I have graphed the points given. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #527631 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39620)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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=527631\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Transverse Axis is the segment connecting the vertices.  Yours is a length of 8 units, so a=4.  Distance from center to either vertex is 4 units, half of the transverse axis length.\r<BR>\n" );
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document.write( "Focal length:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%2Babs%28-5%29%29%2F2=6=c ALIGN=MIDDLE  ALT=\"%287%2Babs%28-5%29%29%2F2=6=c\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Again using focus information, the center, y value is the average of the focus y values:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%2B%28-5%29%29%2F2=1 ALIGN=MIDDLE  ALT=\"%287%2B%28-5%29%29%2F2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The center is (-3,1).\r<BR>\n" );
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document.write( "The negative sign is on the term for x, since your foci are vertically arranged.  If the hyperbola were standard position, vertices would be on the y axis.  In your case, center is NOT at the origin.\r<BR>\n" );
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document.write( "Now you have center, a, and c.  You can say,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28%28y-1%29%5E2%2F4%5E2-%28x%2B3%29%5E2%2Fb%5E2=1%29 ALIGN=MIDDLE  ALT=\"highlight_green%28%28y-1%29%5E2%2F4%5E2-%28x%2B3%29%5E2%2Fb%5E2=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "You still want the b value.\r<BR>\n" );
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document.write( "A substitution is often used in deriving the equation of a hyperbola.  <BR>\n" );
document.write( "The effect is the relationship <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2=c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2=c%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=c%5E2-a%5E2 ALIGN=MIDDLE  ALT=\"b%5E2=c%5E2-a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "You example has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=6%5E2-4%5E2=36-16=20 ALIGN=MIDDLE  ALT=\"b%5E2=6%5E2-4%5E2=36-16=20\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Finally, your standard form hyperbola is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%28y-1%29%5E2%2F4%5E2-%28x%2B3%29%5E2%2F20=1%29 ALIGN=MIDDLE  ALT=\"highlight%28%28y-1%29%5E2%2F4%5E2-%28x%2B3%29%5E2%2F20=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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