If a circle C passing through the point 4 , 0 touches the circle x 2 + y 2 + 4 x - 6 y = 12 externally at the point 1 , - 1 , then the radius of C is:

Step-by-step solution

The equation of the tangent to the circle x 2 + y 2 + 2 g x + 2 f y + c = 0 at a point x 1 ,   y 1 is x x 1 + y y 1 + g x + x 1 + f y + y 1 + c = 0 . Thus, the equation of the tangent to x 2 + y 2 + 4 x - 6 y - 12 = 0 at 1 ,   - 1 is x 1 + y - 1 + 2 x + 1 - 3 y - 1 - 12 = 0 ⇒ 3 x - 4 y - 7 = 0 . Also, we know that the equation of a circle touching a circle x 2 + y 2 + 2 g x + 2 f y + c = 0 and line l x + m y + n = 0 at x 1 ,   y 1 is x - x 1 2 + y - y 1 2 + λ l x + m y + n = 0 Hence, the r