document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=137050>Question 137050</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find an equation of a hyperbola with vertices at (-9,4) and (-5,4) and asymptotes y=3x+25 and y=-3x-17. \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #100296 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20060)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=100296\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find an equation of a hyperbola with vertices at (-9,4) and (-5,4) and asymptotes y=3x+25 and y=-3x-17. \r<BR>\n" );
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document.write( "First let's draw the asymptotes and the vertices:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-12%2C12%2C-8%2C16%2C%0D%0A%0D%0Agraph%28400%2C400%2C-12%2C12%2C-8%2C16%2C0%2C3x%2B25%29%2C%0D%0Agraph%28400%2C400%2C-12%2C12%2C-8%2C16%2C0%2C-3x-17%29%2C+%0D%0Alocate%28-9.2%2C4.6%2Co%29%2Clocate%28-5.25%2C4.6%2Co%29%0D%0A%0D%0A+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "We can tell that the hyperbola opens right and left<BR>\n" );
document.write( "and has the equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "where the center is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28h%2Ck%29%22 ALIGN=MIDDLE  ALT=\"%22%28h%2Ck%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The length of the semi-transverse axis is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The length of the semi-conjugate 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 two asymptotes have slopes ±<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Fa ALIGN=MIDDLE  ALT=\"b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > and pass<BR>\n" );
document.write( "through the center (h,k).\r<BR>\n" );
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document.write( "The center is the midpoint between the vertices,<BR>\n" );
document.write( "(-9,4) and (-5,4) which is the point (-7,4) \r<BR>\n" );
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document.write( "So (h,k) = (-7,4)\r<BR>\n" );
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document.write( "The transverse axis is the distance between the vertices<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28-9%2C4%29%22 ALIGN=MIDDLE  ALT=\"%22%28-9%2C4%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28-5%2C4%29%22 ALIGN=MIDDLE  ALT=\"%22%28-5%2C4%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > units.\r<BR>\n" );
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document.write( "So the semi-transverse axis is half of that or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "That is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since the asymptotes are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%2B25 ALIGN=MIDDLE  ALT=\"y=3x%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-3x-17 ALIGN=MIDDLE  ALT=\"y=-3x-17\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We compare them to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx%2BB ALIGN=MIDDLE  ALT=\"y=mx%2BB\"   DISABLED_style= \"max-width:90%; height:auto;\" > and find that their slopes<BR>\n" );
document.write( "are ±<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,\r<BR>\n" );
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document.write( "Since the slope are ±<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Fa ALIGN=MIDDLE  ALT=\"b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Fa=3 ALIGN=MIDDLE  ALT=\"b%2Fa=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we substitute that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2F2=3 ALIGN=MIDDLE  ALT=\"b%2F2=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=6 ALIGN=MIDDLE  ALT=\"b=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "becomes\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-%28-7%29%29%5E2%2F2%5E2%2B%28y-4%29%5E2%2F6%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-%28-7%29%29%5E2%2F2%5E2%2B%28y-4%29%5E2%2F6%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B7%29%5E2%2F4%2B%28y-4%29%5E2%2F36=1 ALIGN=MIDDLE  ALT=\"%28x%2B7%29%5E2%2F4%2B%28y-4%29%5E2%2F36=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we draw the defining rectangle which has base<BR>\n" );
document.write( "2a and height 2b and has the center as its center.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-12%2C12%2C-8%2C16%2C%0D%0A%0D%0Agraph%28400%2C400%2C-12%2C12%2C-8%2C16%2C0%2C3x%2B25%29%2C%0D%0Agraph%28400%2C400%2C-12%2C12%2C-8%2C16%2C0%2C-3x-17%29%2C+%0D%0Alocate%28-9.2%2C4.6%2Co%29%2Clocate%28-5.25%2C4.6%2Co%29%2C%0D%0Arectangle%28-9%2C-2%2C-5%2C10%29%0D%0A%0D%0A+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Now we can sketch in the hyperbola:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-12%2C12%2C-8%2C16%2C%0D%0A%0D%0Agraph%28400%2C400%2C-12%2C12%2C-8%2C16%2C0%2C3x%2B25%29%2C%0D%0Agraph%28400%2C400%2C-12%2C12%2C-8%2C16%2C0%2C-3x-17%29%2C+%0D%0Alocate%28-9.2%2C4.6%2Co%29%2Clocate%28-5.25%2C4.6%2Co%29%2C%0D%0Arectangle%28-9%2C-2%2C-5%2C10%29%2C%0D%0A%0D%0Agraph%28400%2C400%2C-12%2C12%2C-8%2C16%2C4%2Bsqrt%28-36%2B9%28x%2B7%29%5E2%29%29%2C%0D%0Agraph%28400%2C400%2C-12%2C12%2C-8%2C16%2C4-sqrt%28-36%2B9%28x%2B7%29%5E2%29%29%0D%0A%0D%0A%0D%0A+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
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