ax2 + ay2 + 2gx + 2fy + c = 0....(1)
The abscissa of the points where the circle (1) meets the axis of x i.e. y = 0, are given by the equation
ax2 + 2gx + c = 0 .......(2)
The roots of this equation being x1 and x2 we have
Again, the roots of the equation (2) are both imaginary if g2 < ac. In this case the circle does not meet the axis of x in real points, i.e. geometrically it does not meet the axis of x at all.
The circle will touch the axis of x if the intercept A1A2, be just zero, i.e. ii g2 = ac.
It will meet the axis of x in two points lying on opposite sides of the origin if the two roots of the equation (2) are of opposite signs, i.e. if c be negative.
