Let the abscissae of the two points $$P$$ and $$Q$$ on a circle be the roots of $$x^{2}-4 x-6=0$$ and the ordinates of $$\mathrm{P}$$ and $$\mathrm{Q}$$ be the roots of $$y^{2}+2 y-7=0$$. If $$\mathrm{PQ}$$ is a diameter of the circle $$x^{2}+y^{2}+2 a x+2 b y+c=0$$, then the value of $$(a+b-c)$$ is _____________.

Consider three circles:

$${C_1}:{x^2} + {y^2} = {r^2}$$

$${C_2}:{(x - 1)^2} + {(y - 1)^2} = {r^2}$$

$${C_3}:{(x - 2)^2} + {(y - 1)^2} = {r^2}$$

If a line L : y = mx + c be a common tangent to C_{1}, C_{2} and C_{3} such that C_{1} and C_{3} lie on one side of line L while C_{2} lies on other side, then the value of $$20({r^2} + c)$$ is equal to :

Let a triangle ABC be inscribed in the circle $${x^2} - \sqrt 2 (x + y) + {y^2} = 0$$ such that $$\angle BAC = {\pi \over 2}$$. If the length of side AB is $$\sqrt 2 $$, then the area of the $$\Delta$$ABC is equal to :

Let the tangent to the circle C_{1} : x^{2} + y^{2} = 2 at the point M($$-$$1, 1) intersect the circle C_{2} : (x $$-$$ 3)^{2} + (y $$-$$ 2)^{2} = 5, at two distinct points A and B. If the tangents to C_{2} at the points A and B intersect at N, then the area of the triangle ANB is equal to: