document.write( "Question 601692: Find the equation of the hyperbola whose vertices are at (-1,-5) and
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Algebra.Com's Answer #379954 by Edwin McCravy(20060) $\"\"$ $\"About$
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document.write( "We plot the two vertices (-1,1) and (-1,-5), and draw the transverse\r\n" );
document.write( "axis, \"trans-\" means \"across\" and \"-verse\" means \"vertices\".\r\n" );
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document.write( "The equation of a hyperbola with a vertical transverse axis is\r\n" );
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document.write( " -  = 1\r\n" );
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document.write( "The transverse axis is 6 units long, and the value of \"a\" is the\r\n" );
document.write( "length of the semi-transverse axis, which is half of 6 which is 3.\r\n" );
document.write( "So not we have \r\n" );
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document.write( "a = 3\r\n" );
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document.write( "The center is the midpoint between the two vertices,\r\n" );
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document.write( "which is the point (h,k) = (-1,-2),  So we plot that point:\r\n" );
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document.write( "Next we plot the given focus (-1,7):\r\n" );
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document.write( "The distance known as \"c\" is the distance from the focus to the center.\r\n" );
document.write( "We count tho units and find that it is 5 units fron the center (-1,-2)\r\n" );
document.write( "to the focus (-1,-7).  Therefore c = 5\r\n" );
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document.write( "All hyperbolas have this Pythagorean theorem relationship:\r\n" );
document.write( "Substituting a=3 and c=5 \r\n" );
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document.write( "c² = a² + b²\r\n" );
document.write( "5² = 3² + b²\r\n" );
document.write( "25 = 9 + b²\r\n" );
document.write( "16 = b²\r\n" );
document.write( " 4 = b\r\n" );
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document.write( "Now we have the equation sice we know a = 3, b = 4, (h,k) = (-1,-2)\r\n" );
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document.write( " -  = 1\r\n" );
document.write( "becomes:\r\n" );
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document.write( " -  = 1\r\n" );
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document.write( " -  = 1\r\n" );
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document.write( "That's the answer, but let's finish drawing the graph:\r\n" );
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document.write( "Draw the conjugate axis, which is a horizontal line 2b\r\n" );
document.write( "or 2(4) = 8 units long with the center as its midpoint:\r\n" );
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document.write( "Next we draw the defining rectangle, which is a rectangle with\r\n" );
document.write( "horizontal and vertical sides with the ends of the transverse and\r\n" );
document.write( "conjugate axes as their midpoints:\r\n" );
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document.write( "Draw and extend the diagonals of the defining rectangle:\r\n" );
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document.write( "And finally sketch in the hyperbola with the given vertices and\r\n" );
document.write( "approaching the diagonals:\r\n" );
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document.write( "  \r\n" );
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document.write( "If you were asked to find the equations of the asymptotes it would\r\n" );
document.write( "not be difficult since you have points that each one goes through,\r\n" );
document.write( "the center and corner points of the defining rectangle.\r\n" );
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document.write( "Edwin
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