Two circles, each of radius 5 units, touch each other at (1, 2). If the equation of their common tangent is 4x + 3y = 10, find the equation of the circles.

A circle touches the line y = x at a point P such that OP = $${4\sqrt 2 \,}$$, where O is the origin. The circle contains the point (- 10, 2) in its interior and the length of its chord on the line x + y = 0 is $${6\sqrt 2 \,}$$. Determine the equation of the circle.

If $$\left( {{m_i},{1 \over {{m_i}}}} \right),\,{m_i}\, > \,0,\,i\, = 1,\,2,\,3,\,4$$ are four distinct points on a circle, then show that $${m_1}\,{m_2}\,{m_3}\,{m_4}\, = 1$$

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.