The limiting points of the coaxal system of circles defined...

The limiting points of the coaxal system of circles defined by $x^{2}+y^{2}-6 x-6 y+4=0$ and $x^{2}+y^{2}-2 x-4 y+3=0$ are (a) $(-1,1),\left(\frac{1}{5}, \frac{-8}{5}\right)$ (b) $(1,-1),\left(\frac{1}{5}, \frac{8}{5}\right)$ (c) $(-1,1),\left(\frac{1}{5}, \frac{8}{5}\right)$ (d) None of these

The limiting points of the coaxal system of circles defined by $x^{2}+y^{2}-6 x-6 y+4=0$ and $x^{2}+y^{2}-2 x-4 y+3=0$ are
(a) $(-1,1),\left(\frac{1}{5}, \frac{-8}{5}\right)$
(b) $(1,-1),\left(\frac{1}{5}, \frac{8}{5}\right)$
(c) $(-1,1),\left(\frac{1}{5}, \frac{8}{5}\right)$
(d) None of these

Key Concepts

Coaxial Circles

Coaxial circles are a family of circles that all share a common radical axis or two common limiting points. In such a system, every circle in the family has its center along a fixed line, and they often have a fixed pair of intersection points that remain invariant as the circles vary within the family.

Limiting Points

Limiting points are the fixed points that are common to all circles of a coaxial system. These points do not change as the family of circles is varied and can be real or imaginary. They are obtained by setting conditions that make the circle degenerate or by eliminating the family parameter, thus capturing the intrinsic geometric feature of the circle system.

Radical Axis

The radical axis of two circles is the locus of all points that have equal power with respect to both circles. It is found by subtracting the equations of the circles and plays a crucial role in coaxial systems, as it denotes the common line upon which the centers of the circles typically lie and is instrumental in determining the limiting points.

Equation Manipulation of Circles

Manipulating circle equations—including completing the square, subtracting equations to obtain the radical axis, and parameter elimination—is key in analyzing and solving problems involving coaxial circles and their limiting points. These algebraic techniques help in transitioning from a general equation to a form that clearly reveals the geometric properties of the circles.

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