Circle

The points of intersection of the line \(ax + by = 0,(a \ne b)\) and the circle \({x^2} + {y^2} - 2x = 0\) are \(A(\alpha ,0)\) and \(B(1,\beta )\). The image of the circle with AB as a diameter in the line \(x + y + 2 = 0\) is :