document.write( "Question 822152: What is the standard form of 2y^2+4y=3x^2-6x+9, and what are the foci, vertices, center, and asymptotes? \n" ); document.write( "
Algebra.Com's Answer #494949 by lwsshak3(11628)\"\" \"About 
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What is the standard form of 2y^2+4y=3x^2-6x+9, and what are the foci, vertices, center, and asymptotes?
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\n" ); document.write( "2y^2+4y=3x^2-6x+9
\n" ); document.write( "rearrange terms:
\n" ); document.write( "3x^2-6x-2y^2-4y=-9
\n" ); document.write( "complete the square:
\n" ); document.write( "3(x^2-2x+1)-2(y^2+2y+1)=-9+3-2
\n" ); document.write( "3(x-1)^2-2(y+1)^2=-8
\n" ); document.write( "multiply both sides by -1
\n" ); document.write( "-3(x-1)^2+2(y+1)^2=8
\n" ); document.write( "rearrange terms:
\n" ); document.write( "2(y+1)^2-3(x-1)^2=8
\n" ); document.write( "\"%28y%2B1%29%5E2%2F4-%28x-1%29%5E2%2F%288%2F3%29=1\"
\n" ); document.write( "This is an equation of a hyperbola with vertical transverse axis.
\n" ); document.write( "Its standard form: \"%28y-k%29%5E2%2Fa%5E2-%28x-h%29%5E2%2Fb%5E2=1\"
\n" ); document.write( "For given problem:
\n" ); document.write( "center: (1,-1)
\n" ); document.write( "a^2=4
\n" ); document.write( "a=2
\n" ); document.write( "vertices: (1,-1±a)=(1,-1±2)=(1,-3) and (1,1)
\n" ); document.write( "b^2=8/3
\n" ); document.write( "b=√(8/3)≈√8/√3≈1.633
\n" ); document.write( "c^2=a^2+b^2=4+(8/3)=20/3
\n" ); document.write( "c=√(20/3)≈2.58
\n" ); document.write( "foci: (1,-1±c)=(1,-1±2.58)=(1,-3.58) and (1,1.58)
\n" ); document.write( "..
\n" ); document.write( "Aymptotes are straight lines that go through the center (1,-1)
\n" ); document.write( "Form of equation: y=mx+b, m=slope, b=y-intercept
\n" ); document.write( "slopes of asymptotes for hyperbolas with vertical transverse axis: ±a/b=±2/(√8/√3)=±2√3/√8≈±1.223
\n" ); document.write( "..
\n" ); document.write( "Equation of asymptote with positive slope(1.223):
\n" ); document.write( "y=1.223x+b
\n" ); document.write( "solve for b using coordinates of center
\n" ); document.write( "-1=1.223*1+b
\n" ); document.write( "b=-2.223
\n" ); document.write( "equation: y=1.223x-2.223
\n" ); document.write( "..
\n" ); document.write( "Equation of asymptote with negative slope(-1.223):
\n" ); document.write( "y=-1.223x+b
\n" ); document.write( "solve for b using coordinates of center
\n" ); document.write( "-1=-1.223*1+b
\n" ); document.write( "b=0.223
\n" ); document.write( "equation: y=-1.223x+0.223\r
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