Let $$\left( {x,\,y,\,z} \right)$$ be points with integer coordinates satisfying the system of homogeneous equation: $$$\matrix{ {3x - y - z = 0} \cr { - 3x + z = 0} \cr { - 3x + 2y + z = 0} \cr } $$$

Then the number of such points for which $$^2 + {y^2} + {z^2} \le 100$$ is

The smallest value of $$k$$, for which both the roots of the equation $$${x^2} - 8kx + 16\left( {{k^2} - k + 1} \right) = 0$$$ are real, distinct and have values at least 4, is

The centres of two circles $${C_1}$$ and $${C_2}$$ each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segement joining the centres of $${C_1}$$ and $${C_2}$$ and C a circle touching circles $${C_1}$$ and $${C_2}$$ externally. If a common tangent to $${C_1}$$ and passing through P is also a common tangent to $${C_2}$$ and C, then the radius of the circle C is