The circle $${x^2}\, + \,{y^2} - \,4x\, - 4y + \,4 = 0$$ is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circumcentre of the triangle is $$x\, + \,y\, - xy\, + k\,{\left( {{x^2}\, + \,{y^2}} \right)^{1/2}} = 0$$. Find k.

A polygon of nine sides, each of length $$2$$, is inscribed in a circle. The radius of the circle is .................

In a triangle, the lengths of the two larger sides are $$10$$ and $$9$$, respectively. If the angles are in $$AP$$. Then the length of the third side can be

The set of all $$x$$ for which $$in\left( {1 + x} \right) \le x$$ is equal to ..........