Question:
<p>What is the equation of the circle which passes through the point of intersection of the circles S<sub>1</sub>: x<sup>2</sup> + y<sup>2</sup> - 6x + 2y + 4 = 0, S<sub>2</sub>: x<sup>2</sup> + y<sup>2</sup> + 2x - 4y - 6 = 0 and whose centre lies on the line y = x. </p>
<p style="display:inline">7x<sup>2</sup> + 7y<sup>2</sup> - 8x - 8y - 12 = 0</p>
<p style="display:inline">7x<sup>2</sup> + 7y<sup>2</sup> - 10x - 10y - 12 = 0</p>
<p style="display:inline">x<sup>2</sup> + y<sup>2</sup> + 12x - 12y + 14 = 0</p>
<p style="display:inline">x<sup>2</sup> + y<sup>2</sup> - 6x - 6y + 10 = 0</p>
Step-by-Step Solution
Key Concept: Use the family of circles S₁ + λ(S₁ - S₂) = 0 passing through intersection points, then apply the constraint that center lies on y = x to find λ, ensuring both center coordinates are equal.
<p>7x<sup>2</sup> + 7y<sup>2</sup> - 10x - 10y - 12 = 0</p>
Correct Answer: B