document.write( "Question 740432: I need some help solving a problem. The problem:\r
\n" ); document.write( "\n" ); document.write( "Find the standard form of the hyperbola given below. Find also the vertices, foci, and asymptotes, determine the eccentricity.\r
\n" ); document.write( "\n" ); document.write( "\"+2x%5E2-4y%5E2-4x-8y-18=0+\" \n" ); document.write( "
Algebra.Com's Answer #451627 by lwsshak3(11628)\"\" \"About 
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Find the standard form of the hyperbola given below. Find also the vertices, foci, and asymptotes, determine the eccentricity.\r
\n" ); document.write( "\n" ); document.write( "\"+2x%5E2-4y%5E2-4x-8y-18=0+\"\r
\n" ); document.write( "\n" ); document.write( "complete the square:
\n" ); document.write( "\"+2x%5E2-4x-4y%5E2-8y-18=0+\"
\n" ); document.write( "\"+2%28x%5E2-2x%2B1%29-4%28y%5E2%2B2y%2B1%29=18%2B2-4+\"\"+2%28x-1%29%5E2-4%28y%2B1%29%5E2=16\"
\n" ); document.write( "\"+%28x-1%29%5E2%2F8-%28y%2B1%29%5E2%2F4=1\"
\n" ); document.write( "This is an equation of a hyperbola with horizontal transverse axis.
\n" ); document.write( "Its standard form:\"%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1\", (h,k)=(x,y) coordinates of center
\n" ); document.write( "For given hyperbola:
\n" ); document.write( "center: (1,-1)
\n" ); document.write( "a^2=8
\n" ); document.write( "a=√8≈2.8
\n" ); document.write( "vertices: (1±a,1)=(1±2.8,1)=(1.8,1) and (3.8,1)
\n" ); document.write( "b^2=4
\n" ); document.write( "b=2
\n" ); document.write( "c^2=a^2+b^2=8+4=12
\n" ); document.write( "c=√12≈3.5
\n" ); document.write( "vertices: (1±c,-1)=(1±3.5,-1)=(2.5,-1) and (4.5,-1)
\n" ); document.write( "eccentricity=c/a=√12/√8=√(3/2)≈1.2
\n" ); document.write( "asymptotes: (equations of lines that go thru center of the form: y=mx+b, m=slope, b=y-intercept)
\n" ); document.write( "slopes of asymptotes for hyperbolas with horizontal transverse axis=±b/a=±2/√8
\n" ); document.write( "..
\n" ); document.write( "equation of asymptote with positive slope:
\n" ); document.write( "y=2x/√8+b
\n" ); document.write( "solve for b using coordinates of center:
\n" ); document.write( "-1=2*1/√8+b
\n" ); document.write( "b=-1-2/√8≈1.707
\n" ); document.write( "equation:
\n" ); document.write( "y=2x/√8-1.707
\n" ); document.write( "..
\n" ); document.write( "equation of asymptote with negative slope:
\n" ); document.write( "y=-2x/√8+b
\n" ); document.write( "solve for b using coordinates of center:
\n" ); document.write( "-1=-2*1/√8+b
\n" ); document.write( "b=-1+2/√8≈-0.293
\n" ); document.write( "equation:
\n" ); document.write( "y=-2x/√8-0.293
\n" ); document.write( "..
\n" ); document.write( "see graph below as a visual check:
\n" ); document.write( "y=((x-1)^2/2-4)^.5-1\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
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