Answer
$0$
Work Step by Step
Write the parametric equations as follows:
$ x=2 \\ y=\sqrt 5 \cos \space t \\ z= \sqrt 5\space \sin t $
$ dx=0 \\ dy=-\sqrt 5 \sin t dt \\ dz= \sqrt 5 \space \cos t \space dt $
We need to plug the above values in the given integral.
$$\oint_C F \cdot dr=\int_{0}^{2 \pi} (12 \sqrt 5 \cos t) (0 dt) +(-\sqrt 5 \sin t dt) (9) +(45 \sin^2 t) (\sqrt 5 \cos t dt) \\= \int_{2 \pi}^{0} (-9 \sqrt 5 \sin t+45 \sqrt 5 \sin^2 t \cos t) dt= [9 \sqrt 5 \cos t+\dfrac{45 \sqrt 5}{3} \sin^3 t]_{0}^{2 \pi} \\= (9 \sqrt 5 \cos (2 \pi)+\dfrac{45 \sqrt 5}{3} \sin^3 (2 \pi) )-(9 \sqrt 5 \cos 0+\dfrac{45 \sqrt 5}{3} \sin^3 0) \\ =0$$
