아래 그림과 같이 밑면의 반지름의 길이가 $r$, 높이가 $3r$ 인 원뿔이 다음 조건을 만족시킨다.

(가) 점 $O$ 는 꼭짓점 $A$를 원뿔의 밑면 위로 정사영한 점으로 밑면인 원의 중심이다.
(나) 점 $P'$, $Q'$은 원뿔의 옆면에 있는 점 $P$, $Q$를 각각 밑면 위로 정사영한 점이다.
(다) $\angle P'OQ' = \displaystyle \frac{\pi}{2}$, $\overline{OP'} + \overline{OQ'}=12$
 A$$O$$P'$$Q'$$PQ#gc19118301_3601_0_0{stroke:#ccc;stroke-width:2;stroke-dasharray:5 , 3;fill:none;fill-opacity:1}#gc19118301_3601_0_1{stroke:black;stroke-width:2;stroke-dasharray:none;fill:none}#gc19118301_3601_0_2{stroke:black;stroke-width:.5;stroke-dasharray:none;fill:none}#gc19118301_3601_0_3{stroke:black;stroke-width:.5;stroke-dasharray:5 , 3;fill:none}#gc19118301_3601_0_4{stroke:black;stroke-width:.5;stroke-dasharray:5 , 3;fill:black;fill-opacity:1}#gc19118301_3601_0_5{stroke:black;stroke-width:0;stroke-dasharray:5 , 3;fill:black;fill-opacity:1;font-family:Times;font-size:14px;font-weight:lighter;font-style:normal;text-anchor:middle;alignment-baseline:middle} OP'$Q'$#gc19118301_3601_1_0{stroke:black;stroke-width:0;stroke-dasharray:none;fill:black;fill-opacity:1;font-family:Times;font-size:14px;font-weight:lighter;font-style:normal;text-anchor:middle;alignment-baseline:middle}#gc19118301_3601_1_1{stroke:black;stroke-width:2;stroke-dasharray:none;fill:gray;fill-opacity:.2}#gc19118301_3601_1_2{stroke:black;stroke-width:1;stroke-dasharray:none;fill:none}#gc19118301_3601_1_3{stroke:black;stroke-width:1;stroke-dasharray:none;fill:black;fill-opacity:1}
이때, $\overline{PQ}^{2}$ 의 최솟값을 구하시오. (단, $r > 12$)
A. $90$
B. $36$
C. $92$
D. $134$
E. $72$