Question

When the axes are rotated through an angle \( \theta \) about the origin in the anticlockwise direction and then translated to the new origin (2, -2), if the transformed equation of \( x^2+y^2=4 \) is \( X^2+Y^2+aX+bY+c=0 \), then \( a+b+c= \):

• A. \( 4 \)
• B. \( 8 \)
• C. \( 0 \)
• D. \( 12 \)

Hint:

When transforming equations due to translation, substitute new coordinates directly into the equation and simplify step by step.

Answer 1

The Correct Option is C

The given equation \( x^2 + y^2 = 4 \) represents a circle centered at the origin with radius 2. Applying the transformation: - First, rotation does not affect the equation structurally. - Then, shifting the origin to (2, -2) results in replacing \( x \) and \( y \) with \( (X+2) \) and \( (Y-2) \). Expanding: \[ (X+2)^2 + (Y-2)^2 = 4 \] \[ X^2 + 4X + 4 + Y^2 - 4Y + 4 = 4 \] \[ X^2 + Y^2 + 4X - 4Y + 4 = 4 \] \[ X^2 + Y^2 + 4X - 4Y + 0 = 0 \] Thus, \( a = 4 \), \( b = -4 \), \( c = 0 \), and \( a + b + c = 0 \).