JEE Main & Advanced Mathematics Circle and System of Circles Limiting Points

Limiting points of a system of co-axial circles are the centres of the point circles belonging to the family (Circles whose radii are zero are called point circles).

Let the circle is \[{{x}^{2}}+{{y}^{2}}+2gx+c=0\] …..(i)

where g is a variable and c is a constant.

\[\therefore \]  Centre and the radius of (i) are \[(-g,\,\,0)\] and \[\sqrt{({{g}^{2}}-c)}\] respectively. Let \[\sqrt{{{g}^{2}}-c}=0\]\[\Rightarrow \] \[|g|\ =\ |f|\ =\sqrt{c}=r\]

Thus we get the two limiting points of the given co-axial system as \[{{x}^{2}}+{{y}^{2}}=\frac{{{a}^{2}}}{2}\] and \[(-\sqrt{c},\,\,0)\]

Clearly the above limiting points are real and distinct, real and coincident or imaginary according as \[c>,\,\,=\,\,,<\text{ }0\].