A fixed circle C₁ has the equation x^2 + y^2 - 12x - 16y = 0 . A variable circle C₂ passes through the origin. If one of the diameters of C₂ is a chord of C₁ , and the locus of the centre of C₂ is a circle of radius r , then the value of r^2 is equal to

Step-by-step solution

Let the centre and radius of the fixed circle C₁ be O₁ and R₁ respectively. From the equation x^2 + y^2 - 12x - 16y = 0 , we get: O₁ (6, 8) R₁ = (-6)^2 + (-8)^2 - 0 = 36 + 64 = 10 Let the centre of the variable circle C₂ be O₂(h, k) . Since C₂ passes through the origin (0,0) , its radius R₂ is the distance from (h, k) to the origin: R₂^2 = h^2 + k^2 It is given that one of the diameters of C₂ is a chord of C₁ . This implies that the distance d between the centres O₁ and O₂ forms a right-angled triangle with the rad