Let $$a$$ and $$b$$ be the roots of the equation $${x^2} - 10cx - 11d = 0$$ and those $${x^2} - 10ax - 11b = 0$$ are $$c$$, $$d$$ then the value of $$a + b + c + d,$$ when $$a \ne b \ne c \ne d,$$ is

If $${a_n} = {3 \over 4} - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3} + ....{( - 1)^{n - 1}}{\left( {{3 \over 4}} \right)^n}\,\,and\,\,{b_n} = 1 - {a_n},$$, then find the least natural number $${n_0}$$ such that $${b_n}\,\, > \,\,{a_n}\,\forall \,n\,\, \ge \,\,{n_0}$$.

ABCD is a square of side length 2 units. $$C_1$$ is the circle touching all the sides of the square ABCD and $$C_2$$ is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.

If a circle is such that it touches the line L and the circle $$C_1$$ externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is

ABCD is a square of side length 2 units. $$C_1$$ is the circle touching all the sides of the square ABCD and $$C_2$$ is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point.

A line L' through A is drawn parallel to BD. Point S moves such that its distances from the BD and the vertex A are equal. If locus of S cuts L' at $$T_2$$ and $$T_3$$ and AC at $$T_1$$, then area of $$\Delta \,{T_1}\,{T_2}\,{T_3}$$ is