The locus of point of intersection of the tangents to the circle x^2+y^2=16 , such that the angle between them is 60^ , is

Step-by-step solution

The circle x^2 + y^2 = 16 has center at the origin and radius 4 . For two tangents from an external point P(h, k) intersecting at 60^ , the line from the center to P bisects the angle between the tangents, making the angle between OP and either tangent 30^ . In right triangle OTP , where T is the point of tangency, 30^ = r OP = 4 OP . Since 30^ = 1 2 , we have OP = 8 . The distance OP = h^2 + k^2 = 8 , so squaring both sides yields h^2 + k^2 = 64 . Replacing (h, k) with (x, y) gives the locus x^2 + y^2 = 64 . Final