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The radical axis of two circles S=0, S'=...

The radical axis of two circles S=0, S'=0 does not exist if

A

the centres of the circles are inverse points of S = 0

B

the centres of the circles are conjugate points w.r.t. S = 0

C

the centres of the circles are same

D

the centres are on X-axis.

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To determine when the radical axis of two circles does not exist, we need to understand the concept of the radical axis and the conditions under which it is defined. ### Step-by-Step Solution: 1. **Definition of Radical Axis**: The radical axis of two circles is defined as the locus of points where the lengths of tangents drawn from those points to the two circles are equal. 2. **Conditions for Existence**: The radical axis exists under the following conditions: - If the circles are separate (not touching or overlapping). - If the circles touch each other externally or internally. - If the circles intersect at two points. 3. **Concentric Circles**: The radical axis does not exist when the circles are concentric. Concentric circles are defined as circles that have the same center but different radii. In this case, there is no point from which tangents to both circles can be drawn that are of equal length. 4. **Conclusion**: Therefore, the radical axis of two circles does not exist if the circles are concentric. ### Answer: The correct option is **C**: The centers of the circles are the same (concentric). ---
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