Answer
(a) $conic$ $section$.
(b) $cot\phi =\frac{A-C}{B}$
(c) $B^2-4AC$; $parabola$; an $ellipse$; a $hyperbola$.
Work Step by Step
(a) The general equation $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ can be transformed to $a(x-h)^2+b(Y-k)^2=c$ through coordinate rotations. The graph of the equation is a $conic$ $section$.
(b) Recall the formula used to eliminate the $xy$ term through rotation $angle \phi$ of axes, we have
$cot\phi =\frac{A-C}{B}$ where $A,B,C$ are coefficients of the equation.
(c) Recall the definition of discriminant for the general equation defined above, we have: discriminant=$B^2-4AC$
Depending on the value of the discriminant, the graph will take different shapes, if the discriminant is $0$, the graph is a $parabola$; if it is negative, the graph is an $ellipse$; and if it is positive, the graph is a $hyperbola$.
