Question

Suppose the axes are to be rotated through an angle \( \theta \) so as to remove the \( xy \) term from the equation \(3 x^2 + 2\sqrt{3}xy + y^2 = 0 \). Then in the new coordinate system, the equation \( x^2 + y^2 + 2xy = 2 \) is transformed to:

• A. \( (2 + \sqrt{3})x^2 + (2 - \sqrt{3})y^2 + 2xy = 4 \)
• B. \( (2 - \sqrt{3})x^2 + (2 + \sqrt{3})y^2 - 2xy = 4 \)
• C. \( x^2 + y^2 - 2(2 - \sqrt{3})xy = 4(2 - \sqrt{3}) \)
• D. \( x^2 + y^2 + 2(2 + \sqrt{3})xy = 4(2 + \sqrt{3}) \)

Hint:

To remove the \( xy \)-term in a conic equation, use the rotation of axes technique with the angle \( \theta \) such that \( \tan(2\theta) = \frac{2B}{A - C} \), where \( A \), \( B \), and \( C \) are the coefficients of the quadratic terms.

Answer 1

The Correct Option is A

Step 1: Rotate the coordinate system. We are given the equation \( 3x^2 + 2\sqrt{3}xy + y^2 = 0 \) and we need to remove the \( xy \)-term by rotating the coordinate system. The angle \( \theta \) of rotation is given by: \[ \tan 2\theta = \frac{2B}{A - C} \] where \( A = 3 \), \( B = \sqrt{3} \), and \( C = 1 \). Substituting these values: \[ \tan 2\theta = \frac{2\sqrt{3}}{2} = \sqrt{3}. \] Thus, \( \theta = 45^\circ \). 
Step 2: Apply the transformation. Using the formulas for coordinate rotation, we find the transformed equation: \[ (2 + \sqrt{3})x^2 + (2 - \sqrt{3})y^2 + 2xy = 4. \]