If $${x_1},\,{x_2},\,{x_3}$$ as well as $${y_1},\,{y_2},\,{y_3}$$, are in G.P. with the same common ratio, then the points $$\left( {{x_1},\,{y_1}} \right),\left( {{x_2},\,{y_2}} \right)$$ and $$\left( {{x_3},\,{y_3}} \right).$$

Let $${L_1}$$ be a straight line passing through the origin and $${L_2}$$ be the straight line $$x + y = 1$$. If the intercepts made by the circle $${x^2} + {y^2} - x + 3y = 0$$ on $${L_1}$$ and $${L_2}$$ are equal, then which of the following equations can represent $${L_1}$$?

If two distinct chords, drawn from the point (p, q) on the circle $${x^2}\, + \,{y^2} = \,px\, + \,qy\,\,(\,where\,pq\, \ne \,0)$$ are bisected by the x - axis, then

Let $${T_1}$$, $${T_2}$$ be two tangents drawn from (- 2, 0) onto the circle $$C:{x^2}\,\, + \,{y^2} = 1$$. Determine the circles touching C and having $${T_1}$$, $${T_2}$$ as their pair of tangents. Further, find the equations of all possible common tangents to these circles, when taken two at a time.