In this question my friend,

 

1.    Equation of a tangent at a point (x₁,y₁) { which is on the circle } is                xx₁ + yy₁ = 9 ----------------  Eq 1.

2.    Also, we have x₁² + y₁² = 9 ------------  Eq 2.

3.    in Eq 1,  put x=4, y=3 { as tangent passes through (4,3).

4.    Solve these two equations to get 2 values of  

       x₁ and y₁ each. They represent two points on circle where the two        tangents from point (4,3) touch the circle. Now you have the co-ordinates of 3 points of the triangle and you can find the area easily by using matrix method.