Let A B be a chord of the circle $${x^2} + {y^2} = {r^2}$$ subtending a right angle at the centre. Then the locus of the centriod of the triangle PAB as P moves on the circle is

The equation of the directrix of the parabola $${y^2} + 4y + 4x + 2 = 0$$

The equation of the common tangent touching the circle $${\left( {x - 3} \right)^2} + {y^2} = 9$$ and the parabola $${y^2} = 4x$$ above the $$x$$-axis is

A man from the top of a $$100$$ metres high tower sees a car moving towards the tower at an angle of depression of $${30^ \circ }$$. After some time,the angle of depression becomes $${60^ \circ }$$. The distance (in metres) travelled by the car during this time is