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 C₁, C₂ and C₃ such that C₁ and C₃ lie on one side of line L while C₂ 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 C₁ : x² + y² = 2 at the point M($$-$$1, 1) intersect the circle C₂ : (x $$-$$ 3)² + (y $$-$$ 2)² = 5, at two distinct points A and B. If the tangents to C₂ at the points A and B intersect at N, then the area of the triangle ANB is equal to :

If the tangents drawn at the points $$O(0,0)$$ and $$P\left( {1 + \sqrt 5 ,2} \right)$$ on the circle $${x^2} + {y^2} - 2x - 4y = 0$$ intersect at the point Q, then the area of the triangle OPQ is equal to :