HW$12\mathrm{.}3.$ Impact, pin reaction

A $2\mathrm{.}5-$ kg non$-$uniform slender rod $($length L$=2\mathrm{.}5$m $)$ is at rest in the vertical position shown when it is hit by a

small particle P traveling horizontally. The rod has a fixed pin at its uppermost point O$,$ its radius of gyration

about O is k$\_($O$)=1\mathrm{.}18$m$,$ and its mass center G is located at a distance r$\_($G$)=1$m from O$.$

Find the required location of the impact $($d$)$ so that no horizontal reaction impulse acts at pin O during the

impact.

CORRECT ANSWER: d $=1\mathrm{.}3924$ m

PROVE THIS ANSWER STEP BY STEP OR YOU WILL BE DOWNVOTED. PLEASE.