If the curves, $x^{2}-6 x+y^{2}+8=0$ and $\mathrm{x}^{2}-8 \mathrm{y}+\mathrm{y}^{2}+16-\mathrm{k}=0,(\mathrm{k}>0)$ touch each other at a point, then the largest value of $\mathrm{k}$ is

Let $C_i \equiv  x^2 + y^2 = i^2 (i = 1,2,3)$ are three circles. If there are $4i$ points on circumference of circle $C_i$. If no three of all the points on three circles are collinear then number of triangles which can be formed using these points whose circumcentre does not lie on origin, is-

Consider the circles ${x^2} + {(y - 1)^2} = $ $9,{(x - 1)^2} + {y^2} = 25$. They are such that

The value of $'c'$ for which the set, $\{(x, y) | x^2 + y^2 + 2x \le 1 \} \cap \{(x, y) | x - y + c \ge 0\}$ contains only one point in common is :

If the circles ${x^2} + {y^2} - 2ax + c = 0$ and ${x^2} + {y^2} + 2by + 2\lambda = 0$ intersect orthogonally, then the value of $\lambda $ is

Two circles ${S_1} = {x^2} + {y^2} + 2{g_1}x + 2{f_1}y + {c_1} = 0$ and ${S_2} = {x^2} + {y^2} + 2{g_2}x + 2{f_2}y + {c_2} = 0$ cut each other orthogonally, then