Final answer:
The question involves rotating the axes to transform the equation of a conic section, eliminating the xy-term to simplify it into a standard form. The rotation uses standard coordinate transformation formulas corresponding to the rotation angle of 30°.
Step-by-step explanation:
The problem involves determining the equation of a conic section after a rotation of axes. The equation x² + 2√3xy - y² = 6 represents a conic section in its current form. We must apply a coordinate transformation corresponding to a rotation through an angle Φ = 30° to simplify the equation and eliminate the xy-term.
To transform the coordinates, we use the rotation formulas:
x' = x cos(Φ) - y sin(Φ)
y' = x sin(Φ) + y cos(Φ)
where x' and y' are the new coordinates after rotation.
By substituting these expressions into the original equation and simplifying, we would obtain the transformed equation of the conic in the x'y'-coordinate system. This process will yield a conic in standard form without the xy-term.