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A circle S₁ has the equation x^2 + y^2 - 2x - 4y - 4 = 0 . A second circle S₂ has a radius of 5 , and its centre lies on the line x + y = 7 . If one of the diameters of S₁ is a chord of S₂ , then the sum of the squares of the x -coordinates of the possible centres of S₂ is

Correct answer

26

Step-by-step solution

Let the centre and radius of the circle S₁ be C₁ and R₁ respectively. From the given equation x^2 + y^2 - 2x - 4y - 4 = 0 , we have: C₁ (1, 2) R₁ = 1^2 + 2^2 - (-4) = 9 = 3 Let the centre of the circle S₂ be C₂ . Since it lies on the line x + y = 7 , we can assume C₂ (h, 7 - h) . The radius of S₂ is given as R₂ = 5 . It is given that one of the diameters of S₁ is a chord of S₂ . This means the distance d between the centres C₁ and C₂ satisfies the right-angled triangle condition with the radii: d^2 + R₁^2 = R₂^2 Su

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