If we rewrite the general equation of degree 2 (Eq. 12) in terms of variables tildex and tildey that are related to x and y by Eqs. (13) and (14), we obtain a new equation of degree 2 in tildex and tildey of the same form but with different coefficients: a^ prime tildex²+b^ prime tildex tildey+c^ prime tildey²+d^ prime tildex+e^ prime tildey+f^ prime=0. (a) Show that b^ prime=b cos 2θ+(c-a) sin 2θ. (b) Show that if b neq 0, then we obtain b^ prime=0 forθ= frac12 cot ⁻¹ fraca-cb This proves that it is always possible to eliminate the cross term b x y by rotating the axes through a suitable angle.
