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 $${m_1}\,{m_2}\,{m_3}\,{m_4}\, = 1$$

Let a given line $$L_1$$ intersects the x and y axes at P and Q, respectively. Let another line $$L_2$$, perpendicular to $$L_1$$, cut the x and y axes at R and S, respectively. Show that the locus of the point of intersection of the lines PS and QR is a circle passing through the origin.