Answer:
see below
Step-by-step explanation:
It can be useful to remember that the angle θ of rotation between the x-y plane and the x'-y' plane is given by ...
cot(2θ) = (A -C)/B
where the standard form equation is ...
Ax² +Bxy +Cy² +Dx +Ey +F = 0
For a 30° rotation, you want ...
(A -C)/B = cot(2·30°) = 1/√3
Looking at the answer choices, you have ...
A: (A -C)/B = (3 -(-39))/(42√3) = 1/√3 . . . as required
B: (A -C)/B = (3 -(-39))/(-12√3) = 7/(2√3)
C: (A -C)/B = (-39 -3)/(42√3) = -1/√3
D: AC > 0, not a hyperbola
_____
Based on rotation angle only, the appropriate choice is the first choice. Because the term signs in the x'-y' formula are different, the figure is a hyperbola. In general form, the equation describes a hyperbola when AC<0.
