Let C₁ be a circle in the first quadrant of radius r > 2 , that touches both the coordinate axes. Let C₂ be a circle that touches C₁ externally at the point (2,1) . If the centre of C₂ lies on the line x = -4 , and the area of C₂ is k , then the value of k is :

Step-by-step solution

Let the centre of C₁ be (r, r) since it touches both axes in the first quadrant. The point P(2,1) lies on C₁ . (2-r)^2 + (1-r)^2 = r^2 r^2 - 6r + 5 = 0 (r-1)(r-5) = 0 Since r > 2 , we have r = 5 . The centre of C₁ is A(5,5) . Since C₂ touches C₁ externally at P(2,1) , the centres of C₁ and C₂ and the point of contact P are collinear. The line passing through A(5,5) and P(2,1) has slope m = 5-1 5-2 = 4 3 . Equation of the line joining the centres is y - 1 = 4 3 (x - 2) 4x - 3y - 5 = 0 . The centre of C₂ , say B , li