If the tangent at the point $$P$$ on the circle $${x^2} + {y^2} + 6x + 6y = 2$$     meets a straight line $$5x -2y + 6 =0$$    at a point $$Q$$ on the $$y$$-axis, then the length of $$PQ$$  is-

Solution :
Line $$5x - 2y + 6 = 0$$    is intersected by tangent at $$P$$ to circle $${x^2} + {y^2} + 6x + 6y - 2 = 0$$      on $$y$$-axis at $$Q\,\left( {0,\,3} \right).$$
In other words tangent passes through $$\left( {0,\,3} \right)$$
$$\therefore \,\,PQ = $$     length of tangent to circle from $$\left( {0,\,3} \right)$$
$$\eqalign{ & = \sqrt {0 + 9 + 0 + 18 - 2} \cr & = \sqrt {25} \cr & = 5 \cr} $$