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.

$$C_1$$ and $$C_2$$ are two concentric circles, the radius of $$C_2$$ being twice that of $$C_1$$. From a point P on $$C_2$$, tangents PA and PB are drawn to $$C_1$$. Prove that the centroid of the triangle PAB lies on $$C_1$$.

Let C be any circle with centre $$\,\left( {0\, , \sqrt {2} } \right)$$. Prove that at the most two rational points can to there on C. (A rational point is a point both of whose coordinates are rational numbers.)

A circle passes through three points A, B and C with the line segment AC as its diameter. A line passing through A angles DAB and CAB are $$\,\alpha \,\,and\,\,\beta $$ respectively and the distance between the point A and the mid point of the line segment DC is d, prove that the area of the circle is $$${{\pi \,{d^2}\,\,{{\cos }^2}\,\,\alpha } \over {{{\cos }^2}\,\alpha \, + \,{{\cos }^2}\,\beta \, + \,\,2\,\cos \,\,\alpha \,\,\cos \,\beta \,\cos \,\,(\beta - \alpha )\,}}$$$