A system of circles is said to be coaxial when every pair of the circles has the same…

A system of circles is said to be coaxial when every pair of the circles has the same radical axis. It follows from this definition that :
1 . The centres of all circles of a coaxial system lie on one straight line, which is perpendicular to the common radical axis.
2 . Circles passing through two fixed points form a coaxial system for which the line joining the fixed points is the common radical axis.
3 . The equation to a coaxial system, of which two members are S₁ = 0 and S₂ = 0, is S₁ + λS₂ = 0, λ is parameter. If we choose the line of centres as x–axis and the common radical axis as y–axis, then the simplest form of equation of coaxial circles is x² + y² + 2gx + c = 0 ...(1)
where c is fixed and g is arbitrary.
If g = ± \(\sqrt{\mathrm{c}},\) then the radius \(\sqrt{\mathrm{g}^{2}-\mathrm{c}}\) vanishes and the circles become point circles. The points (± \(\sqrt{\mathrm{c}},\)0) are called the limiting points of the system of coaxial circles given by (1).
If origin be a limiting point of a coaxial system one of whose member is x² + y² – 2αx – 2βy + c = 0, then the other limiting point is