The angle between the tangents to the circle with centre $(4,5) $ drawn from $P(-2,-3) $ is $120^o$ , the length of the tangent to the circle from $P$ is :

Distance from P(-2,-3) to center (4,5) is d = sqrt((4 - -2)^2 + (5 - -3)^2) = sqrt(6^2 + 8^2) = 10. The angle between tangents is 120, so the angle between the line connecting the center to P and the tangent is 60. Let r be the radius. sin(60) = r/d = r/10. r = 10 * sqrt(3)/2 = 5*sqrt(3). The length of the tangent L = sqrt(d^2 - r^2) = sqrt(100 - 75) = sqrt(25) = 5.