How do you identify the following equation # 9x^2 - 3y^2 = 27# as a circle, parabola, ellipse or hyperbola?

Firstly, to see whether it is a parabola, we have to note whether only one kind of variable has the square term with it. i.e #y^2=4ax" or "x^2=4ay#. Since the given equation has 2 power terms we conclude that the equation doesn't trace a parabola.

Next, to identify if the given equation is a circle, one has to check if the coefficients of the square powered variables are equal.
i.e they follow the equation #ax^2+bx^2=c" where "a=b#
This particular equation does not have its square coefficients equal and hence we are safe to say that it is not a circle.

Since we ruled out on whether it is a circle or parabola, we need to look at the power terms now. Now check the signs to see whether it is an ellipse or hyperbola.

If the given equation the variables grouped up such that #(x-l)^2/a+(y-m)^2/b=1# then we can continue. Since in this equation there is no such requirement we can continue without any extra-thoughts.

Now, to check whether it is a hyperbola or an ellipse, check the signs the variables carry. If any one of the variables has a minus sign beside it, then the equation is a hyperbola. In the given equation, we see the #y# variable has the negative sign, and hence we conclude that it is an equation for a hyperbola.

Note, if there were no negative signs, then the equation would be that of an ellipse.