document.write( "Question 458612: One of the asymptotes of a hyperbola with a vertical transverse axis is y-4 = 2/3(x-1) The distance between the two vertices is 8 units. What is the equation of the hyperbola and what are the coordinates of the two foci? \n" ); document.write( "
Algebra.Com's Answer #314944 by lwsshak3(11628)\"\" \"About 
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One of the asymptotes of a hyperbola with a vertical transverse axis is y-4 = 2/3(x-1) The distance between the two vertices is 8 units. What is the equation of the hyperbola and what are the coordinates of the two foci
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\n" ); document.write( "Standard form of hyperbola with vertical transverse axis: (y-h)^2/a^2-(x-h)^2/b^2=1, with (h,k) being the (x,y) coordinates of the center.
\n" ); document.write( "length of vertical transverse axis=8=2a
\n" ); document.write( "a=4
\n" ); document.write( "a^2=16
\n" ); document.write( "..
\n" ); document.write( "asymptotes:
\n" ); document.write( "y-4 = 2/3(x-1)
\n" ); document.write( "y=(2/3)x-2/3+4
\n" ); document.write( "y=(2/3)x-2/3+12/3
\n" ); document.write( "y=(2/3)x+10/3 (given asymptote)
\n" ); document.write( "and
\n" ); document.write( "y=-(2/3)x+10/3 (other asymptote)
\n" ); document.write( "y-intercept=10/3=y-coordinate of center
\n" ); document.write( "x-coordinate of center=0
\n" ); document.write( "..
\n" ); document.write( "slope,m=2/3=a/b
\n" ); document.write( "b=(3/2)a=(3/2)4=6
\n" ); document.write( "b^2=36
\n" ); document.write( "Equation: (y-10/3)^2/16-x^2/36=1
\n" ); document.write( "..
\n" ); document.write( "Foci:
\n" ); document.write( "c^2=a^2+b^2=16+36=52
\n" ); document.write( "c=√52=7.21..
\n" ); document.write( "coordinates of Foci:(0,10/3±√52)
\n" ); document.write( "see graph below as a visual check on the answers\r
\n" ); document.write( "\n" ); document.write( "..
\n" ); document.write( "y=(16+16x^2/36)^.5+10/3
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