A circle C₁ lies in the first quadrant, touches the x-axis at the point (3,0) , and makes an intercept of length 8 on the y-axis. Let C₂ be a circle with centre at (11, 20) and radius r . If C₁ and C₂ intersect at exactly two distinct points for all r ( , ) , then the value of 2 - is equal to :

Step-by-step solution

Let the centre of C₁ be (3, R) . Since it touches the x-axis, its radius is R . The length of the y-intercept is given by 2 R^2 - 3^2 = 8 . Squaring both sides, we get R^2 - 9 = 16 R^2 = 25 R = 5 (since C₁ is in the first quadrant). Thus, C₁ has its centre at (3,5) and radius r₁ = 5 . The centre of C₂ is (11, 20) and its radius is r₂ = r . The distance d between the centres is: d = (11-3)^2 + (20-5)^2 = 8^2 + 15^2 = 17 For two circles to intersect at exactly two distinct points, we must have: |r₁ - r₂| |r - 5| From