Root locus is always symmetric about which axis?

• A. Imaginary axis
• B. Real axis
• C. Vertical axis
• D. Origin

Answer: (B)

Root locus plots are graphical representations used in control systems to show how the roots of a system's characteristic equation change as a particular parameter (commonly a gain) varies. The important aspect of root locus is its symmetry properties, which directly relate to the complex nature of the roots.

The correct answer, which states that the root locus is always symmetric about the real axis, is based on the principle that complex roots of polynomials with real coefficients come in conjugate pairs. This means if there is a root at a point in the complex plane, there will always be another root at the mirrored position across the real axis. For instance, if a system has a root at a complex number ( s = a + bi ), there will be a corresponding root at ( s = a - bi ). This inherent symmetry ensures that as one traces the path of the root locus, the plot will reflect this conjugate nature, resulting in a symmetrical appearance with respect to the real axis.

In contrast, the imaginary axis or any other options do not possess this symmetry in the context of polynomial roots. The imaginary axis does not guarantee conjugate pairing for roots, as a root could exist solely in the imaginary plane without its conjugate counterpart, which implies no necessary