Consider points s1 = -3 + j4 & s2 =  -3 - j2 in the s-plane for a system withG(s)H(s) = Select one:a. s1 is on root locus but not s2b. s2 is on root locus but not s1c. Neither s1 nor s2 is on root locusd. both s1 and s2 is on root locus

To determine whether points s1 = -3 + j4 and s2 = -3 - j2 are on the root locus, we need to analyze the system with G(s)H(s).

The root locus is a plot of the possible locations of the poles of a system as a parameter (usually a gain) varies. It helps us understand how the system's stability changes with different parameter values.

To determine if a point is on the root locus, we need to check if it satisfies the root locus rules. These rules state that the sum of the angles of the vectors from the poles to the point should be an odd multiple of 180 degrees, and the number of poles and zeros to the right of the point should be equal.

Let's analyze each point:

For s1 = -3 + j4:

• The angle of the vector from the pole to s1 is atan(4/(-3)) = -53.13 degrees.
• Since there are no poles or zeros to the right of s1, the number of poles and zeros to the right is 0.
• The sum of the angles is -53.13 degrees, which is not an odd multiple of 180 degrees.

For s2 = -3 - j2:

• The angle of the vector from the pole to s2 is atan((-2)/(-3)) = 33.69 degrees.
• Since there are no poles or zeros to the right of s2, the number of poles and zeros to the right is 0.
• The sum of the angles is 33.69 degrees, which is not an odd multiple of 180 degrees.

Based on the analysis, neither s1 nor s2 satisfies the root locus rules. Therefore, the correct answer is c. Neither s1 nor s2 is on the root locus.