document.write( "Question 482542: Explain which conic section this equation and explain how to solve it:
\n" ); document.write( "12x^2-18y^2-18x-12y+12=0 \n" ); document.write( "
Algebra.Com's Answer #332942 by lwsshak3(11628)\"\" \"About 
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Explain which conic section this equation and explain how to solve it:
\n" ); document.write( "12x^2-18y^2-18x-12y+12=0
\n" ); document.write( "***
\n" ); document.write( "12x^2-18y^2-18x-12y+12=0
\n" ); document.write( "dividing by 3 to simplify
\n" ); document.write( "3x^2-6y^2-6x-4y+4=0
\n" ); document.write( "completing the square
\n" ); document.write( "3(x^2-2x+1)-6(y^2+(2/3)y+1/9)=-4+3-2/3=-1-2/3=-5/3
\n" ); document.write( "3(x-1)^2-6(y-1/3)^2=-5/3
\n" ); document.write( "multiply both sides by (-1)
\n" ); document.write( "6(y-1/3)^2-3(x-1)^2=5/3
\n" ); document.write( "divide both sides by 5/3
\n" ); document.write( "(y-1/3)^2/(5/18)-(x-1)^2/(5/9)=1
\n" ); document.write( "This is an equation of a hyperbola with vertical transverse axis of the standard form:
\n" ); document.write( "(y-k)^2/a^2-(x-h)^2/b^2=1
\n" ); document.write( "For given equation:
\n" ); document.write( "Center:(1,1/3)
\n" ); document.write( "a^2=5/18
\n" ); document.write( "a=√(5/18)
\n" ); document.write( "Length of transverse axis=2a=2*√(5/18)
\n" ); document.write( "Vertices: (1,1/3±a), (1,1/3±√(5/18))
\n" ); document.write( "b^2=5/9
\n" ); document.write( "b=√(5/9)
\n" ); document.write( "Length of conjugate axis=2b=2*√(5/9)
\n" ); document.write( "c^2=a^2+b^2=5/18+5/9=15/18
\n" ); document.write( "c=√(15/18)
\n" ); document.write( "Foci: (1,1/3±c)=(1,1/3±√(15/18))
\n" ); document.write( "Slope of asymptotes=±a/b=±√(5/18)/√(5/9)=±√(1/2)=±√2/2
\n" ); document.write( "Equation of asymptotes:
\n" ); document.write( "Asymptotes are straight lines of the standard form: y=mx+b, m=slope, b=y-intercept
\n" ); document.write( "Asymptote with positive slope:
\n" ); document.write( "y=(√2/2)x+b
\n" ); document.write( "solving for b, using center coordinates thru which asymptotes pass
\n" ); document.write( "1/3=(√2/2)*1+b
\n" ); document.write( "b=-.3737
\n" ); document.write( "equation: y=.707x-.3737
\n" ); document.write( "..
\n" ); document.write( "Asymptote with negative slope:
\n" ); document.write( "Equation: y=-.707x+1.0403
\n" ); document.write( "..
\n" ); document.write( "Comment: As you can see, there is a whole bunch of things that can be asked about a hyperbola. \n" ); document.write( "
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