A point on the real axis is part of the root locus if the number of poles and zeros to its right is:

• A. Even
• B. Zero
• C. Odd
• D. Infinite

Answer: (C)

In root locus analysis, the placement of poles and zeros on the complex plane is crucial for determining the behavior of a control system as feedback gain changes. According to the root locus rules, a point on the real axis is included in the root locus if the number of poles and zeros to its right is odd.

When assessing a point on the real axis, counting the poles (which are typically represented by 'X') and the zeros (represented by 'O') to the right of the point is essential. If that count is odd, it implies a change in the number of roots; hence, this point will be part of the root locus.

This is fundamentally rooted in the principle of how the poles and zeros affect the trajectory of the system's poles as gain varies. An odd count results in a sign change in the angle criterion applied to the closed-loop transfer function, providing a valid path in the root locus plot. Thus, the significance of having an odd number of poles and zeros to the right of a point confirms its inclusion in the root locus, which is foundational for understanding the stability and dynamic response of control systems.