If a diameter of the circle x^2 + y^2 - 4x + 6y + k = 0 is a chord of the circle x^2 + y^2 - 10x - 6y - 15 = 0 , then the value of k is

Step-by-step solution

Let the first circle be S₁ x^2 + y^2 - 4x + 6y + k = 0 . Its center is C₁(2, -3) and its radius is R₁ = 2^2 + (-3)^2 - k = 13 - k . Let the second circle be S₂ x^2 + y^2 - 10x - 6y - 15 = 0 . Its center is C₂(5, 3) and its radius is R₂ = 5^2 + 3^2 - (-15) = 25 + 9 + 15 = 7 . The distance d between the centers C₁ and C₂ is given by: d^2 = (5 - 2)^2 + (3 - (-3))^2 = 3^2 + 6^2 = 9 + 36 = 45 . Since a diameter of S₁ is a chord of S₂ , the center C₁ lies on the chord of S₂ . The distance from the center C₂ to this chord