Question

The equation of a circle which passes through the points of intersection of the circles \[ 2x^2 + 2y^2 - 2x + 6y - 3 = 0, \quad x^2 + y^2 + 4x + 2y + 1 = 0 \] and whose centre lies on the common chord of these circles is: 

• A. \( 2x^2 + 2y^2 - 3x + 4y - 2 = 0 \)
• B. \( x^2 + y^2 + 2x + 5y - 2 = 0 \)
• C. \( 3x^2 + 3y^2 - 2x + 4y - 3 = 0 \)
• D. \( 4x^2 + 4y^2 + 6x + 10y - 1 = 0 \)

Hint:

The equation of a circle passing through the intersection of two given circles is obtained using the relation \( S_1 + \lambda S_2 = 0 \). The center condition helps determine the correct value of \( \lambda \).

Answer 1

The Correct Option is D

Step 1: Finding the Equation of the Common Chord 
The given circles are: \[ 2x^2 + 2y^2 - 2x + 6y - 3 = 0, \] \[ x^2 + y^2 + 4x + 2y + 1 = 0. \] Subtracting the second equation from the first: \[ (2x^2 + 2y^2 - 2x + 6y - 3) - (x^2 + y^2 + 4x + 2y + 1) = 0. \] \[ x^2 + y^2 - 6x + 4y - 4 = 0. \] 

Step 2: Finding the Required Circle 
A circle passing through the intersection of these two given circles is of the form: \[ S_1 + \lambda S_2 = 0. \] Substituting and simplifying using the condition that the center lies on the common chord, we derive: \[ 4x^2 + 4y^2 + 6x + 10y - 1 = 0. \] 

Step 3: Conclusion 
Thus, the final answer is: \[ \boxed{4x^2 + 4y^2 + 6x + 10y - 1 = 0}. \]