G(S)H(S)=$\frac{K(S+4)(S+5)}{(S+3)(S+1)}$

Here, no. of poles P=2

No. of zeros Z=2

No. of branches ending at ∞,

N=P-Z=2-2

=0

Angle of Asymptoe

Θ=$\frac{(2q+1)180}{(P-Z)}$, q=0, 1,…….,(P-Z-1)

As P-Z=0, Angle of asymptote is not applicable

Centroid -> NA

The characteristic equaton is,

1+G(S)H(S)=0

$\frac{1+k(S^2+9S+20)}{(S^2+4S+3)}$=0

$\frac{k(S^2+9S+20)}{(S^2+4S+3)}$=-1

k=$\frac{-(S^2+4S+3)}{(S^2+9S+20)}$

$\frac{dk}{ds}$=0

-$\frac{(S^2+4S+3)(2S+9)-(S^2+9S+20)(2S+4)}{(S^2+9S+20)^2]}$=0

$(S^2+4S+3)(2S+9)-(S^2+9S+20)(2S+4)$=0

$2S^3+8S^2+6S+9S^2+36S+27-(2S^3+4S^2+18S^2+36S+40S+80)$=0

$(17S^2-22S^2)+42S-76S+27-80$=0

$-5S^2-34S-53$=0

$5S^2+34S+53$=0

S=$\frac{-3±\sqrt{(34^2-4 x S x 53)}}{(2 x S)}$

S=-4.79, -2.241

As both the values of ‘S’ lies on root locus. Both are valid break away points.

Point of intersection with imaginary axis,

$S^2$ (1+k) (20k+3)

$S^1$ (4+9k) 0

$S^0$ (20k+3) 0

(1+k) ≥ 0 k_max=-1

(4+9k) ≥ 0 k_max=(-4)/9

(20k+3) ≥ 0 k_max=(-3)/20

There is no point of intersection with imaginary axis.