Consider the region R = {(x, y) $$\in$$ R $$\times$$ R : x $$\ge$$ 0 and y² $$\le$$ 4 $$-$$ x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circles in F. Let ($$\alpha$$, $$\beta$$) be a point where the circle C meets the curve y² = 4 $$-$$ x.

The value of $$\alpha$$ is ___________.

Let O be the centre of the circle x² + y² = r², where $$r > {{\sqrt 5 } \over 2}$$. Suppose PQ is a chord of this circle and the equation of the line passing through P and Q is 2x + 4y = 5. If the centre of the circumcircle of the triangle OPQ lies on the line x + 2y = 4, then the value of r is .............

Let the point B be the reflection of the point A(2, 3) with respect to the line $$8x - 6y - 23 = 0$$. Let $$\Gamma_{A} $$ and $$\Gamma_{B} $$ be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles $$\Gamma_{A} $$ and $$\Gamma_{B} $$ such that both the circles are on the same side of T. If C is the point of intersection of T and the line passing through A and B, then the length of the line segment AC is .................

The straight line 2x - 3y = 1 divides the circular region $${x^2}\, + \,{y^2}\, \le \,6$$ into two parts.
If $$S = \left\{ {\left( {2,\,{3 \over 4}} \right),\,\left( {{5 \over 2},\,{3 \over 4}} \right),\,\left( {{1 \over 4} - \,{1 \over 4}} \right),\,\left( {{1 \over 8},\,{1 \over 4}} \right)} \right\}$$ then the number of points (s) in S lying inside the smaller part is