document.write( "Question 759881: what is the standard form of the equation of a hyperbola with vertex (0,4) focus at (0,5) and center at (0,1)? \n" ); document.write( "

Algebra.Com's Answer #462298 by Edwin McCravy(20060) $\"\"$ $\"About$

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what is the standard form of the equation of a hyperbola with vertex (0,4) focus at (0,5) and center at (0,1)?
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document.write( "Hyperbolas have the equation \r\n" );
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document.write( "if they look like this: )(\r\n" );
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document.write( "and the equation\r\n" );
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document.write( "if they look like this:\r\n" );
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document.write( "We plot the three given points for the vertex V, focus F \r\n" );
document.write( "and center C:\r\n" );
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document.write( "So it looks like this:\r\n" );
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document.write( "and has the equation\r\n" );
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document.write( "We know that the center (h,k) is (0,1). We know that \r\n" );
document.write( "the semi-transverse axis, a, is the distance from the \r\n" );
document.write( "center to a vertex, and it is 3 units from C to V, so\r\n" );
document.write( "a=3.  There is another vertex 3 units below the center\r\n" );
document.write( "at (0,-2).\r\n" );
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document.write( "So we now have everything but b:\r\n" );
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document.write( "We know that c is the distance from the center to a \r\n" );
document.write( "focus, and there are 4 units from C to F so c=4.  \r\n" );
document.write( "There is another focus 4 units below the center at \r\n" );
document.write( "(0,-3).\r\n" );
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document.write( "In all hyperbolas we have the Pythagorean property\r\n" );
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document.write( "c² = a² + b²\r\n" );
document.write( "4² = 3² + b²\r\n" );
document.write( "16 = 9 + b²\r\n" );
document.write( " 7 = b²\r\n" );
document.write( "√7 = b\r\n" );
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document.write( "So now we know that b² = 7, a² =3² = 9, so the equation is:\r\n" );
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document.write( "or change (x-0)² to just x²\r\n" );
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document.write( "Here is the complete graph.  The conjugate axis is the horizontal\r\n" );
document.write( "line through the center, the width of the defining rectangle.\r\n" );
document.write( "It is 2b units wide or 2√7, √7 on each side or about 2.7 on each \r\n" );
document.write( "side of the center.\r\n" );
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document.write( "Edwin

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