The angle between a pair of tangents drawn from a point P to the circle $${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,9\,{\sin ^2}\,\alpha \, + \,13\,{\cos ^2}\,\alpha \, = \,0$$ is $$2\,\alpha $$.
The equation of the locus of the point P is

The circles $${x^2} + {y^2} - 10x + 16 = 0$$ and $${x^2} + {y^2} = {r^2}$$ intersect each other in two distinct points if

The locus of the centre of a circle, which touches externally the circle $${x^2} + {y^2} - 6x - 6y + 14 = 0$$ and also touches the y-axis, is given by the equation:

The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle $${x^2} + {y^2} = 9$$is