Answer
(a) hyperbola.
(b) $\phi=\frac{\pi}{4}$, $\frac{Y^2}{8}-\frac{X^2}{8}=1$
(c) See graph.
Work Step by Step
(a) Rewrite the original equation as $0x^2+xy+0y^2+0x+0y+4=0$ and we have $A=C=D=E=0, B=1, F=4$. The discriminant is $B^2-4AC=1$, thus the graph of the equation is a hyperbola.
(b) To eliminate the xy-term, we need a rotation of axes of angle $\phi$ 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 $\frac{\sqrt 2}{2}(X-Y)\frac{\sqrt 2}{2}(X+Y)+4=0$ or
$X^2-Y^2+8=0$ or $\frac{Y^2}{8}-\frac{X^2}{8}=1$
(c) See graph.
