Let $$M = \{ (x,y) \in R \times R:{x^2} + {y^2} \le {r^2}\} $$, where r > 0. Consider the geometric progression $${a_n} = {1 \over {{2^{n - 1}}}}$$, n = 1, 2, 3, ...... . Let S₀ = 0 and for n $$\ge$$ 1, let S_n denote the sum of the first n terms of this progression. For n $$\ge$$ 1, let C_n denote the circle with center (S_n$$-$$1, 0) and radius a_n, and D_n denote the circle with center (S_n$$-$$1, S_n$$-$$1) and radius a_n.

Consider M with $$r = {{1025} \over {513}}$$. Let k be the number of all those circles C_n that are inside M. Let l be the maximum possible number of circles among these k circles such that no two circles intersect. Then

Let $$M = \{ (x,y) \in R \times R:{x^2} + {y^2} \le {r^2}\} $$, where r > 0. Consider the geometric progression $${a_n} = {1 \over {{2^{n - 1}}}}$$, n = 1, 2, 3, ...... . Let S₀ = 0 and for n $$\ge$$ 1, let S_n denote the sum of the first n terms of this progression. For n $$\ge$$ 1, let C_n denote the circle with center (S_n$$-$$1, 0) and radius a_n, and D_n denote the circle with center (S_n$$-$$1, S_n$$-$$1) and radius a_n.

Consider M with $$r = {{({2^{199}} - 1)\sqrt 2 } \over {{2^{198}}}}$$. The number of all those circles D_n that are inside M is

Consider a triangle $$\Delta$$ whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocenter of $$\Delta$$ is (1, 1), then the equation of the circle passing through the vertices of the triangle $$\Delta$$ is

A line y = mx + 1 intersects the circle $${(x - 3)^2} + {(y + 2)^2}$$ = 25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate $$ - {3 \over 5}$$, then which one of the following options is correct?