I Need Help Writing an Equation of a Hyperbola Without an 'xy' term.

I Need Help Writing an Equation of a Hyperbola Without an 'xy' term.

Can somebody please walk me through (step by step) how I write an equation for this hyperbola without an xy term.

xy - x - y - 1 = 0

The answer is (1/2)x^2 - (1/2)y^2 - ∫2x - 1 = 0

How do you do this???

Useful link:
http://www.stewartcalculus.com/data/CALC…

Here, A = 0, B = 1, C = 0.

To eliminate the xy-term, choose t such that cot(2t) = (A - C)/B.
==> cot(2t) = 0
==> 2t = π/2
==> t = π/4 (or 45 degrees).

So, we let x = u cos t - v sin t and y = u sin t + v cos t
(I'm using u and v instead of X and Y to minimize confusion.)

Here, we have
x = u cos(π/4) - v sin(π/4) = (u - v)/√2
y = u sin(π/4) + v cos(π/4) = (u + v)/√2

Substitute this into the original equation:
[(u - v)/√2][(u + v)/√2] - (u - v)/√2 - (u + v)/√2 - 1 = 0
==> (u^2 - v^2)/2 - u√2 - 1 = 0
==> u^2 - v^2 - 2u√2 - 2 = 0.

I hope this helps!