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 $$${{\pi \,{d^2}\,\,{{\cos }^2}\,\,\alpha } \over {{{\cos }^2}\,\alpha \, + \,{{\cos }^2}\,\beta \, + \,\,2\,\cos \,\,\alpha \,\,\cos \,\beta \,\cos \,\,(\beta - \alpha )\,}}$$$

Consider a family of circles passing through two fixed points A (3, 7) and B (6, 5). Show that the chords on which the circle $${x^2}\, + \,{y^2} - \,4x - \,6y - 3 = 0$$ cuts the members of the family are concurrent at a point. Find the coordinate of this point.

Find the coordinates of the point at which the circles $${x^2}\, + \,{y^2} - \,4x - \,2y = - 4\,\,and\,\,{x^2}\, + \,{y^2} - \,12x - \,8y = - 36$$ touch each other. Also find equations common tangests touching the circles in the distinct points.

Let a circle be given by 2x (x - a) + y (2y - b) = 0, $$(a\, \ne \,0,\,\,b\, \ne 0)$$. Find the condition on a abd b if two chords, each bisected by the x-axis, can be drawn to the circle from $$\left( {a,\,\,{b \over 2}} \right)$$.