document.write( "Question 251077: Two condominium apartment buildings that face each other have a front door awning that is shaped like a hyperbola. The equation of the hyperbola is 25x^2 - 81y^2 = 30,625. Find the distance in meters between the two front door awnings at their closest point. \n" ); document.write( "
Algebra.Com's Answer #182884 by jsmallt9(3758)\"\" \"About 
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Here are the keys to solving this:
  • The points where the two parts of a hyperbola are closest are the vertices.
  • The center of a hyperbola is halfway between the vertices.
  • The distance from the center of a hyperbola to one of its veritices is \"a\".
  • Since the center is halfway between the vertices and the distance from the center to a vertex is \"a\", then the distance between the two vertices is 2a.

\n" ); document.write( "So all we have to do to solve the problem is figure out what \"a\" is and then multiply it by 2.

\n" ); document.write( "We can find \"a\" is we write the equation for the hyperbola in the form:
\n" ); document.write( "\"%28x-h%29%5E2%2Fa%5E2+-+%28y-k%29%5E2%2Fb%5E2+=+1\"
\n" ); document.write( "So we want to transform \"25x%5E2+-+81y%5E2+=+++30625\" into the form above. We can start by getting the 1 on the right by dividing both sides by 30625:
\n" ); document.write( "\"25x%5E2%2F30625+-+81y%5E2%2F30625+=+1\"
\n" ); document.write( "Next we can get rid of the coefficients of each fraction by multiplying the numerator and denominator of each fraction by the reciprocal of its coefficient:
\n" ); document.write( "
\n" ); document.write( "which gives us:
\n" ); document.write( "\"x%5E2%2F%2830625%2F25%29+-+y%5E2%2F%2830625%2F81%29+=+1\"
\n" ); document.write( "which simplifies to
\n" ); document.write( "\"x%5E2%2F1225+-+y%5E2%2F%2830625%2F81%29+=+1\"
\n" ); document.write( "Since x = x-0 and y = y-0 we can rewrite this as:
\n" ); document.write( "\"%28x-0%29%5E2%2F1225+-+%28y-0%29%5E2%2F%2830625%2F81%29+=+1\"
\n" ); document.write( "And we have the desired form with \"a%5E2+=+1225\". This makes a = 35 and the distance between the vertices, which is 2a, 70. \n" ); document.write( "
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