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In the co-axial system of circle ${x^2} + {y^2} + 2gx + c = 0$, where $g$ is a parameter, if $c > 0$ then the circles are
Orthogonal
Touching type
Intersecting type
Non-intersecting type
Solution
(b) Given, equation of the circle $x^2 + y^2 + 2gx + c = 0$
where $c$ is constant and $g$ represents the parameter of a coaxial system and $c > 0.$
We know that the standard equation of a circle is ${x^2} + {y^2} + 2gx + 2fy + c = 0.$
Comparing the given equation with the standard equation, we get centre $ \equiv ( – g,\,0)$ and radius $\sqrt {{g^2} – c} $.
Therefore radius becomes zero, when ${g^2} – c = 0$ or $g = \pm \sqrt c .$
Therefore $(\sqrt c ,\,0)$ and $( – \sqrt c ,\,0)$ are the limiting points of the coaxial system of circles.
Since $c > 0$, therefore $\sqrt c $ is real and limiting points are real and distinct.
Thus the co-axial system is said to be touching type.
