$$\because$$ $$p + q = 3$$ ...... (i)

and $${p^4} + {q^4} = 369$$ ...... (ii)

$${\{ {(p + q)^2} - 2pq\} ^2} - 2{p^2}{q^2} = 369$$

or $${(9 - 2pq)^2} - 2{(pq)^2} = 369$$

or $${(pq)^2} - 18pq - 144 = 0$$

$$\therefore$$ $$pq = - 6$$ or 24

But $$pq = 24$$ is not possible

$$\therefore$$ $$pq = - 6$$

Hence, $${\left( {{1 \over p} + {1 \over q}} \right)^{ - 2}} = {\left( {{{pq} \over {p + q}}} \right)^2} = {( - 2)^2} = 4$$