There is a line $l$ passing through the origin O with slope $\tan \theta$. Let $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ be the feet of the perpendiculars from two points $\mathrm { A } ( 0,2 ) , \mathrm { B } ( 2 \sqrt { 3 } , 0 )$ to line $l$, respectively. What is the value of $\theta$ that maximizes the sum of the distances from the origin O to point $\mathrm { A } ^ { \prime }$ and to point $\mathrm { B } ^ { \prime }$, $\overline { \mathrm { OA } ^ { \prime } } + \overline { \mathrm { OB } ^ { \prime } }$? (Given that $0 < \theta < \frac { \pi } { 2 }$.) [3 points] (1) $\frac { \pi } { 12 }$ (2) $\frac { \pi } { 6 }$ (3) $\frac { \pi } { 4 }$ (4) $\frac { \pi } { 3 }$ (5) $\frac { 5 } { 12 } \pi$
There is a line $l$ passing through the origin O with slope $\tan \theta$. Let $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ be the feet of the perpendiculars from two points $\mathrm { A } ( 0,2 ) , \mathrm { B } ( 2 \sqrt { 3 } , 0 )$ to line $l$, respectively. What is the value of $\theta$ that maximizes the sum of the distances from the origin O to point $\mathrm { A } ^ { \prime }$ and to point $\mathrm { B } ^ { \prime }$, $\overline { \mathrm { OA } ^ { \prime } } + \overline { \mathrm { OB } ^ { \prime } }$? (Given that $0 < \theta < \frac { \pi } { 2 }$.) [3 points]\\
(1) $\frac { \pi } { 12 }$\\
(2) $\frac { \pi } { 6 }$\\
(3) $\frac { \pi } { 4 }$\\
(4) $\frac { \pi } { 3 }$\\
(5) $\frac { 5 } { 12 } \pi$