To identify a 2D conic, why translate ax^2+by^2+2hxy+2gx+2fy=0 to remove the x and y term first?

Question

I heard if we remove the x and y term first by a translation, then it would be easier for us use another rotation to get a standard equation of a conic, e.g. hyperbola, parabola or ellipse, as it would save us from doing completing squares.
However, I don't understand why a translation to remove x and y terms helps. What is the underlying working or deduction suporting this?
What I think is that based on what I heard, for any ellipse, hyperbola or parabola, as long as they are in their standard positions, then no matter how much or how many times they rotate, there will not be any x or y term.
Well, this works for ellipse and hyperbola, as the rotation is performed by putting x = x ' cos@ - y ' sin@ and y = x ' sin@ + y ' cos@, after substituting, there will just be x^2, y^2, xy and a constant term. There is no x or y term.
But for an ellipse, y^2=4ax, after putting x, there will be x and y terms.
My question is whether what I have just said is correct and how the translation helps?

Answer 1

y^2 = 4ax is a parabola, not a hyperbola.

Removing the xy term reduces the equation to the standard form which can then be easily compared and the curve can be deduced.

Imagine programming a computer to identify the curve, which would be easier ?

Elimination of xy or completion of squares ?