Hi;

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.

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

is

We want this to vanish, so any

such that

and

will do. So we take

. Hence

Thus under the transformation the curve

becomes the hyperbola

. Furthermore as the transformation

represents 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.