Two questions about rotation of this line

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The discussion revolves around the rotation of the conic section represented by the equation 3x^2 - 2√3xy + y^2 = 1. A discriminant test indicates that the equation describes lines rather than a parabola, which raises questions about the nature of the conic. The formula for cotangent of the rotation angle suggests that a negative angle could imply a clockwise rotation, leading to confusion about the rotation direction. Participants note that the original poster has not clearly stated the specific problems they are facing. Clarification on the problem statement is needed to provide more focused assistance.
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Two questions about rotation of this "line"

Homework Statement




3x^2 - 2\sqrt{3}xy +y^2 = 1

Homework Equations



x = x'\cos\alpha - y'\sin\alpha
y = x'\sin\alpha + y'\cos\alpha


The Attempt at a Solution



The discriminant test is

4(3) - 4(3)(1) = 0

Supposed to be a parabola, instead we get lines.

One other thing

Supposed that in

cot2\alpha = \frac{A - C}{B}

And then we get that \alpha < 0, can that happen? Can we rotate clockwise?
 
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The OP is missing a problem statement. What is it that you are having problems with?
 

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