Let the circles C₁ : |z| = r and C₂ : |z - 3 - 4i| = 5 , z C , be such that C₂ lies within C₁ . If z₁ moves on C₁ , z₂ moves on C₂ and |z₁ - z₂| = 2 , then |z₁ - z₂| is equal to:

Step-by-step solution

The center of circle C₁ is O₁(0,0) and its radius is r₁ = r . The center of circle C₂ is O₂(3,4) and its radius is r₂ = 5 . The distance between the centers of the two circles is d = 3^2 + 4^2 = 5 . Since C₂ lies entirely within C₁ , the minimum distance between a point on C₁ and a point on C₂ is given by r₁ - (d + r₂) . We are given that |z₁ - z₂| = 2 . r - (5 + 5) = 2 r - 10 = 2 r = 12 The maximum distance between a point on C₁ and a point on C₂ is given by r₁ + d + r₂ . |z₁ - z₂| = 12 + 5 + 5 = 22 Answer: 22