In questions of this sort I tend look for a transformation to get rid of the term:

When this is substituted into the original equation, the term in

isWe want this to vanish, so *any*

Thus under the transformation the curve

becomes the hyperbola . Furthermore as the transformationrepresents a clockwise rotation of 45° about the origin followed by an enlargement of

at the origin, the conic section is preserved, i.e. the original curve is indeed a hyperbola.**NB: Be careful when using linear transformations on curves: only rotations, reflections and enlargements/contractions by a nonzero factor preserve conic sections. Any other transformation may distort the curve and alter its original nature.**

Like the light sabre by the way. You beat me to it.

Agnishom: Here's my version:

Substitute* x = X +1 and y = Y + 1, where X and Y are new variables.

So we now have a more familiar XY = 0 (the rectangular hyperbola)

Now substitute* X = x/a - y/b and Y = x/a + y/b

* substitutions like these preserve the hyperbolic nature of the curve.

Bob

]]>He can put it in standard form with a rotation of 45 degrees clockwise and a translation by the √2

Just involves the substitutions

of

and then replacing x1 by x1+ √2.

You might download this

http://math.sci.ccny.cuny.edu/document/show/2685

rename the file to Rotation of Axes.pdf This will explain some of this, won't make you as good as scientia or bob bundy with these transformation problems but it is a start.

]]>You mean as in the one given here: wiki?

Put h = k = m = 1, and you have this equation.