A circle C₁ has its centre on the line y = x in the first quadrant and touches the x-axis. Let C₂ be a circle with its centre at (0, k) , where k > 0 , and radius r . If the set of all values of r for which C₁ and C₂ intersect at exactly two distinct points is the interval (3, 11) , then the value of (k - 4)^2 is equal to :

Step-by-step solution

Let the centre of C₁ be (a, a) with a > 0 . Since C₁ touches the x-axis, its radius is r₁ = a . The centre of C₂ is (0, k) and its radius is r₂ = r . The distance d between the centres of C₁ and C₂ is: d = (a - 0)^2 + (a - k)^2 = a^2 + (k - a)^2 For the two circles to intersect at exactly two distinct points, the radius r must satisfy: |r - r₁| This compound inequality can be rewritten to isolate r as: d - r₁ r₁ , which is consistent with the given positive interval bounds). We are given that this interval is (3, 1