document.write( "Question 1043381: Find the standard equation of the hyperbola with the center at the origin, vertices at (+- 10, 0) and asymptotes y = +- (3/4)x \n" ); document.write( "

Algebra.Com's Answer #658535 by Edwin McCravy(20060) $\"\"$ $\"About$

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document.write( "Hyperbolas that look like this--->   )( \r\n" );
document.write( "have equation:\r\n" );
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document.write( "Where (h,k) is the center, a is the length of the semi-transverse\r\n" );
document.write( "axis, and b is the length of the semi-conjugate axis.\r\n" );
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document.write( "The center is the origin so (h,k) = (0,0),\r\n" );
document.write( "The semi-transverse axis is the line from the center to the\r\n" );
document.write( "vertex.  This is 10 units long, so a = 10.  So we substitute\r\n" );
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document.write( "We only need b, the length of the semi-conjugate axis.\r\n" );
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document.write( "The asymptotes are the extended diagonals of the defining rectangle.\r\n" );
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document.write( "The defining rectangle of a hyperbola is the rectangle such that\r\n" );
document.write( "1. Its dimensions are 2a by 2b, the lengths of the transverse and\r\n" );
document.write( "conjugate axes.\r\n" );
document.write( "2. Its diagonals intersect at the center of the hyperbola.\r\n" );
document.write( "3. The midpoints of two opposite sides of the rectangle are the\r\n" );
document.write( "vertices of the hyperbola.\r\n" );
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document.write( "The upper right corner of the defining rectangle is the point where\r\n" );
document.write( "the vertical line x=10 intersects the asymptote with the positive\r\n" );
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document.write( "To find the point which is the upper right corner of the defining\r\n" );
document.write( "rectangle, we solve the system\r\n" );
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document.write( "and get the point \r\n" );
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document.write( "So the length of the semi-conjugate axis is .\r\n" );
document.write( "That is the value of b.  So we substitute b=\r\n" );
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document.write( "That's the standard equation.\r\n" );
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document.write( "Edwin

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