Answer
(a) Hyperbola. (b) $\phi=\frac{\pi}{4}$. $X^2-Y^2=16$ (c) See graph.
Work Step by Step
(a) Given the equation $xy=8$ or $0x^2+xy+0y^2+0x+0y-8=0$, we have $A=0, B=1, C=0, D=0, E=0, F=8$. The discriminant is $B^2-4AC=1\gt0$, thus the equation represents a hyperbola.
(b) To eliminate the xy-term, we do a rotation of angle $\phi$ of the axes where $cot2\phi=\frac{A-C}{B}=0$ which gives $2\phi=\frac{\pi}{2}$ and $\phi=\frac{\pi}{4}$. The transformation formula gives $x=X\cdot cos\frac{\pi}{4} - Y\cdot sin\frac{\pi}{4}=\frac{\sqrt 2}{2}(X-Y)$, $y=X\cdot sin\frac{\pi}{4}+Y\cdot cos\frac{\pi}{4}=\frac{\sqrt 2}{2}(X+Y)$
Use them in the original equation to get $\frac{\sqrt 2}{2}(X-Y)\frac{\sqrt 2}{2}(X+Y)=8$ or $X^2-Y^2=16$
(c) See graph.
