Question

The origin is shifted to the point \( (2, 3) \) by translation of axes and then the coordinate axes are rotated about the origin through an angle \( \theta \) in the counter-clockwise sense. Due to this if the equation \( 3x^2 + 2xy + 3y^2 - 18x - 22y + 50 = 0 \) is transformed to \( 4x^2 + 2y^2 - 1 = 0 \), then the angle \( \theta = \):

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{6} \)
• D. \( \frac{\pi}{2} \)

Hint:

When rotating axes, use the standard rotation transformation formulas and simplify the equation accordingly.

Answer 1

The Correct Option is A

We are given the equation \( 3x^2 + 2xy + 3y^2
- 18x
- 22y + 50 = 0 \) and its transformation to \( 4x^2 + 2y^2
- 1 = 0 \). The rotation of axes is performed to simplify the equation.
The general form for a rotated equation is: \[ x' = x \cos \theta
- y \sin \theta, \quad y' = x \sin \theta + y \cos \theta. \] By applying these rotation formulas to the given equation and comparing it to the transformed equation \( 4x^2 + 2y^2
- 1 = 0 \), we find the angle \( \theta \) to be \( \frac{\pi}{4} \).