For the co-axial system
x2 + y2 + 2λx + c = 0
i) the radical axis is the Y-axis (i.e, x = 0)
ii) the line of centres is the X-axis (i.e, y = 0)
iii) the two points of intersection of radical axis and each circle are
P (0, + √(–c)) and Q(0, –√(–c)).
Now 'c' can be less than zero or equal to zero or greater than zero.
a) c < 0, i.e, √(–c) is real.
⇒ The points of intersection P and Q are real.
Such a system is called intersecting co-axial system. Refer to figure at the top.
b) c = 0, i.e, P and Q coincide.
⇒ The radical axis touches each circle of the co-axial system.
Such a system is called touching co-axial system. Three such systems are shown below.
c) c > 0, i.e, ± √(–c) is imaginary.
The radical axis does not meet any of the circles.
Such a system is called non-intersecting co-axial system. Refer to figure at the bottom.
The members of a co-axial system of circles with zero radius are defined as it's limiting points.
The limiting points of x2 + y2 + 2λx + c = 0 are (+ √c, 0) and (– √c, 0).
i) if c > 0, the limiting points exist and are distinct
ii) if c = 0, the limiting points exist and coincide
iii) if c < 0, the limiting points do not exist