Question

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Answer 1

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.