Two questions about rotation of this line

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SUMMARY

The discussion focuses on the rotation of the conic section defined by the equation 3x² - 2√3xy + y² = 1. The discriminant test reveals that the equation represents lines rather than a parabola, as indicated by the calculation yielding a discriminant of zero. Additionally, the discussion raises a question about the angle of rotation, specifically whether a negative angle (α < 0) indicates a clockwise rotation, which is a valid consideration in the context of conic section transformations.

PREREQUISITES
  • Understanding of conic sections and their equations
  • Familiarity with rotation transformations in coordinate geometry
  • Knowledge of the discriminant test for conic sections
  • Basic trigonometry, specifically related to angles and rotation
NEXT STEPS
  • Study the properties of conic sections, focusing on parabolas and lines
  • Learn about the discriminant method for classifying conic sections
  • Research rotation transformations in coordinate geometry, including the implications of negative angles
  • Explore the derivation and application of the cotangent formula for angle of rotation in conics
USEFUL FOR

Students and educators in mathematics, particularly those studying conic sections, geometry, and transformations. This discussion is also beneficial for anyone looking to deepen their understanding of the relationship between algebraic equations and geometric interpretations.

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Two questions about rotation of this "line"

Homework Statement




[tex]3x^2 - 2\sqrt{3}xy +y^2 = 1[/tex]

Homework Equations



[tex]x = x'\cos\alpha - y'\sin\alpha[/tex]
[tex]y = x'\sin\alpha + y'\cos\alpha[/tex]


The Attempt at a Solution



The discriminant test is

[tex]4(3) - 4(3)(1) = 0[/tex]

Supposed to be a parabola, instead we get lines.

One other thing

Supposed that in

[tex]cot2\alpha = \frac{A - C}{B}[/tex]

And then we get that [tex]\alpha < 0[/tex], 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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