Let C₁ be a circle of radius 5 in the first quadrant that touches both the coordinate axes. Let C₂ be a circle of radius 10 that touches C₁ externally. If the centre of C₂ lies on the line 4x - 3y - 5 = 0 in the first quadrant, and L is the common tangent to C₁ and C₂ at their point of contact, then the area of the triangle formed by L and the coordinate axes is :

Step-by-step solution

Let the centre of C₁ be A . Since C₁ touches both axes in the first quadrant and has radius 5, A is (5,5) . The centre of C₂ , say B , lies on the line 4x - 3y - 5 = 0 . Notice that the coordinates of A(5,5) satisfy this equation ( 4(5) - 3(5) - 5 = 0 ). Thus, the line joining the centres A and B is exactly 4x - 3y - 5 = 0 . Since C₁ and C₂ touch externally, the distance between their centres is AB = r₁ + r₂ = 5 + 10 = 15 . The slope of the line AB is 4 3 , so = 3 5 and = 4 5 . The coordinates of B are (5 15 3 5 ,