For two positive numbers $k , p$, two tangent lines are drawn from point $\mathrm { A } ( - k , 0 )$ to the parabola $y ^ { 2 } = 4 p x$. Let $\mathrm { F } , \mathrm { F } ^ { \prime }$ be the two points where these tangent lines meet the $y$-axis, and let $\mathrm { P } , \mathrm { Q }$ be the two points where they meet the parabola. When $\angle \mathrm { PAQ } = \frac { \pi } { 3 }$, if the length of the major axis of the ellipse with foci at $\mathrm { F } , \mathrm { F } ^ { \prime }$ and passing through points $\mathrm { P } , \mathrm { Q }$ is $4 \sqrt { 3 } + 12$, what is the value of $k + p$? [4 points] (1) 8 (2) 10 (3) 12 (4) 14 (5) 16
For two positive numbers $k , p$, two tangent lines are drawn from point $\mathrm { A } ( - k , 0 )$ to the parabola $y ^ { 2 } = 4 p x$. Let $\mathrm { F } , \mathrm { F } ^ { \prime }$ be the two points where these tangent lines meet the $y$-axis, and let $\mathrm { P } , \mathrm { Q }$ be the two points where they meet the parabola. When $\angle \mathrm { PAQ } = \frac { \pi } { 3 }$,\\
if the length of the major axis of the ellipse with foci at $\mathrm { F } , \mathrm { F } ^ { \prime }$ and passing through points $\mathrm { P } , \mathrm { Q }$ is $4 \sqrt { 3 } + 12$, what is the value of $k + p$? [4 points]\\
(1) 8\\
(2) 10\\
(3) 12\\
(4) 14\\
(5) 16