A hyperbola has asymptotes with equations $y = \pm \frac { 4 } { 3 } x$ and two foci at $\mathrm { F } ( c , 0 )$, $\mathrm { F } ^ { \prime } ( - c , 0 )$ $(c > 0)$, and satisfies the following conditions. (a) For a point P on the hyperbola, $\overline { \mathrm { PF } ^ { \prime } } = 30$ and $16 \leq \overline { \mathrm { PF } } \leq 20$. (b) For the vertex A with positive $x$-coordinate, the length of segment AF is a natural number. Find the length of the major axis of this hyperbola. [4 points]
A hyperbola has asymptotes with equations $y = \pm \frac { 4 } { 3 } x$ and two foci at $\mathrm { F } ( c , 0 )$, $\mathrm { F } ^ { \prime } ( - c , 0 )$ $(c > 0)$, and satisfies the following conditions.\\
(a) For a point P on the hyperbola, $\overline { \mathrm { PF } ^ { \prime } } = 30$ and $16 \leq \overline { \mathrm { PF } } \leq 20$.\\
(b) For the vertex A with positive $x$-coordinate, the length of segment AF is a natural number.\\
Find the length of the major axis of this hyperbola. [4 points]