Correct Answer - Option 4 : 3 and (-4, 0)

**Concept:**

Number of asymptotes = |P – Z|

P = number of open-loop poles

Z = number of open-loop zeros

Centroid:

It is the intersection of the asymptotes and always lies on the real axis. **It is denoted by σ.**

\(σ = \frac{{\sum {P_i} - \sum {Z_i}}}{{\left| {P - Z} \right|}}\)

ΣPi is the sum of real parts of finite poles of G(s)H(s)

ΣZi is the sum of real parts of finite zeros of G(s)H(s)

**Calculation:**

given that s(s + 4) (s + 5) (s + 6) + K(s + 3) = 0

characteristic equation 1 + G(s)H(s) = 0

\(1 + \frac{{K\left( {s + 3} \right)}}{{s\left( {s + 4} \right)\left( {s + 5} \right)\left( {s + 6} \right)}}=0\)

\(G(s)H(s)= \frac{{K\left( {s + 3} \right)}}{{s\left( {s + 4} \right)\left( {s + 5} \right)\left( {s + 6} \right)}}\)

from the above transfer function

P = 4 and Z = 1

**1) Number of asymptotes = |P – Z| = |4 - 1| = 3**

**2) Centroid**

**\(σ = \frac{{\sum {P_i} - \sum {Z_i}}}{{\left| {P - Z} \right|}}\)**

\(σ = \frac{{\sum {(0 -4-5-6)} - \sum {(-3)}}}{{\left| {4-1} \right|}}\)

**σ = -12 / 3 = -4**

**so number of asymptotes and centroid are -3 and (-4, 0**)