Let $$PQ$$ be a focal chord of the parabola $${y^2} = 4ax$$. The tangents to the parabola at $$P$$ and $$Q$$ meet at a point lying on the line $$y=2x+a$$, $$a>0$$.

If chord $$PQ$$ subtends an angle $$\theta $$ at the vertex of $${y^2} = 4ax$$, then tan $$\theta = $$

Let $$PQ$$ be a focal chord of the parabola $${y^2} = 4ax$$. The tangents to the parabola at $$P$$ and $$Q$$ meet at a point lying on the line $$y=2x+a$$, $$a>0$$.

Length of chord $$PQ$$ is

A line $$L:y=mx+3$$ meets $$y$$-axis at R$$(0, 3)$$ and the arc of the parabola $${y^2} = 16x,$$ $$0 \le y \le 6$$ at the point $$F\left( {{x_0},{y_0}} \right)$$. The tangent to the parabola at $$F\left( {{x_0},{y_0}} \right)$$ intersects the $$y$$-axis at $$G\left( {0,{y_1}} \right)$$. The slope $$m$$ of the line $$L$$ is chosen such that the area of the triangle $$EFG$$ has a local maximum.

Match List $$I$$ with List $$II$$ and select the correct answer using the code given below the lists:

List $$I$$
P.$$\,\,\,m = $$
Q.$$\,\,\,$$Maximum area of $$\Delta EFG$$ is
R.$$\,\,\,$$ $${y_0} = $$
S.$$\,\,\,$$ $${y_1} = $$

List $$II$$
1.$$\,\,\,$$ $${1 \over 2}$$
2.$$\,\,\,$$ $$4$$
3.$$\,\,\,$$ $$2$$
4.$$\,\,\,$$ $$1$$

Circle (s) touching x-axis at a distance 3 from the origin and having an intercept of length $$2\sqrt 7 $$ on y-axis is (are)