document.write( "Question 1042705: Find the equation of a hyperbola satisfying the given conditions.
\n" ); document.write( "Asymptotes y=1/2x, y=-1/2x; one vertex(4,0) \n" ); document.write( "

Algebra.Com's Answer #657746 by KMST(5328) $\"\"$ $\"About$

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The asymptotes of a hyperbola intersect at the center of a hyperbola,
\n" ); document.write( "so this hyberbola is centered at the origin,
\n" ); document.write( "and that makes life easier.
\n" ); document.write( "The equation of a hyperbola centered at the origin is
\n" ); document.write( "of the form $\"x%5E2%2Fa%5E2-y%5E2%2Fb%5E2=1\"$ , where it is possible to have $\"y=0\"$ ,
\n" ); document.write( "or of the form $\"y%5E2%2Fa%5E2-x%5E2%2Fb%5E2=1\"$ , where it is not possible to have $\"y=0\"$ .
\n" ); document.write( "(In that equation $\"a\"$ and $\"b\"$ are real numbers,
\n" ); document.write( "and they are positive by customary definition).
\n" ); document.write( "Since vertex (4,0) with $\"y=0\"$ is part of the hyperbola,
\n" ); document.write( "this hyperbola has an equation of the form $\"x%5E2%2Fa%5E2-y%5E2%2Fb%5E2=1\"$ .
\n" ); document.write( "Now, al we have to do is find $\"a\"$ and $\"b\"$ .
\n" ); document.write( "Substituting the coordinates for the given vertex into $\"x%5E2%2Fa%5E2-y%5E2%2Fb%5E2=1\"$ , we see that
\n" ); document.write( " $\"4%5E2%2Fa%5E2-0%5E2%2Fb%5E2=1\"$ --> $\"16%2Fa%5E2=1\"$ ---> $\"a%5E2=16\"$ --> $\"a=4\"$ .
\n" ); document.write( "So far, we have {x^2/16- .
\n" ); document.write( "As we go farther and farther from the center along the hyperbola,
\n" ); document.write( "the terms $\"x%5E2%2F4%5E2\"$ and $\"y%5E2%2Fb%5E2\"$ to the left of the equal sign
\n" ); document.write( "get so large compared to the $\"1\"$ on the other side,
\n" ); document.write( "that the hyperbola can be approximated by
\n" ); document.write( " $\"x%5E2%2F16=y%5E2%2Fb%5E2\"$ <---> $\"y%5E2=b%5E2y%5E2%2F4%5E2\"$ <--> $\"system%28y=bx%2F4%2C%22or%22%2Cy=-bx%2F4%29\"$ .
\n" ); document.write( "Since far from the center the hyperbola \"can be approximated\" by its asymptotes,
\n" ); document.write( "the last two linear equations abovemust be the equations of the asymptotes.
\n" ); document.write( "Since the equations given for the asymptotes,
\n" ); document.write( " $\"y=%281%2F2%29x\"$ and $\"y=-%281%2F2%29x\"$ ,
\n" ); document.write( "must be the same as $\"y=%28b%2F4%29x\"$ and $\"y=%28b%2F4%29x\"$ ,
\n" ); document.write( " $\"b%2F4=1%2F2\"$ ---> $\"b=2\"$ .
\n" ); document.write( "Putting it all together, tHe equation of the hyperbola is
\n" ); document.write( " $\"x%5E2%2F16-y%5E2%2F2%5E2=1\"$ or $\"highlight%28x%5E2%2F16-y%5E2%2F4=1%29\"$ . \n" ); document.write( "

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