the general form of the equation of a hyperbola is:
(x-h)^2 / a^2 - (y-k)^2 / b^2 = 1
your equation is 3x^2 - 2y^2 = 6
divide both sides of this equation by 6 to get:
3x^2/6 - 2y^2/6 = 1
simplify to get:
x^2/2 - y^2/3 = 1
this means that a^2 = 2 and b^2 = 3
this means that a = sqrt(2) and b = sqrt(3)
the equation for the asymptotes of a horizontally aligned hyperbola is:
y = plus or minus (b/a)*(x-h) + k
since h and k are equal to 0, this equation becomes y = plus or minus (b/a)x.
since b = sqrt(3) and a = sqrt(2), this becomes y = plus or minus sqrt(3)/sqrt(2) * x.
since sqrt(3) / sqrt(2) is the same as sqrt(3/2), this becomes y = plus or minus sqrt(3/2) * x.
the equation of the asymptotes is y = plus or minus sqrt(3/2)x.
the following graph shows the equation of the hyperbola and the equations of the asymptotes.
all 6 of the blue equations point to the same hyperbola on the graph.
those equations are equivalent to each other.
the last 2 of the 6 go together.
one of them is plus the square root of and the other is minus the square root of.
the red equations are the equations of the asymptotes of the hyperbola.
one of them is plus the equation and the other is minus the equation.
read the reference.
there's lots of good information in there and it's not that hard to follow.
