Two tangents are drawn from a point P to the circle x 2 + y 2 - 2 x - 4 y + 4 = 0 , such that the angle between these tangents is tan - 1 12 5 , where tan - 1 12 5 ∈ ( 0 , π ) . If the centre of the circle is denoted by C and these tangents touch the circle at points A and B , then the ratio of the areas of Δ P A B and Δ C A B is :

Step-by-step solution

Let, the angle between the tangents P A and P B from the point P to the circle x 2 + y 2 - 2 x - 4 y + 4 = 0 with centre C is θ . Given θ = tan - 1 12 5     ⇒ tan θ = 12 5 . We know that the centre and radius of a circle x 2 + y 2 + 2 g x + 2 f y + c = 0 are respectively - g ,   - f and g 2 + f 2 - c . Thus, the centre and radius of the circle x 2 + y 2 - 2 x - 4 y + 4 = 0 are respectively 1 ,   2 and 1 2 + 2 2 - 4 = 1 unit. We know that the line joining the centre of a circ