As shown in the figure, an ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 25 } = 1$ has foci at $\mathrm { F } ( 0 , c ) , \mathrm { F } ^ { \prime } ( 0 , - c )$. Let A be the point with positive $x$-coordinate where the ellipse meets the $x$-axis. Let B be the intersection of the line $y = c$ and the line $\mathrm { AF } ^ { \prime}$, and let P be the point with positive $x$-coordinate where the line $y = c$ meets the ellipse. If the difference between the perimeter of triangle $\mathrm { BPF } ^ { \prime}$ and the perimeter of triangle BFA is 4, what is the area of triangle $\mathrm { AFF } ^ { \prime}$? (Given: $0 < a < 5 , c > 0$) [3 points] (1) $3 \sqrt { 6 }$ (2) $\frac { 7 \sqrt { 6 } } { 2 }$ (3) $4 \sqrt { 6 }$ (4) $\frac { 9 \sqrt { 6 } } { 2 }$ (5) $5 \sqrt { 6 }$
As shown in the figure, an ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 25 } = 1$ has foci at $\mathrm { F } ( 0 , c ) , \mathrm { F } ^ { \prime } ( 0 , - c )$. Let A be the point with positive $x$-coordinate where the ellipse meets the $x$-axis. Let B be the intersection of the line $y = c$ and the line $\mathrm { AF } ^ { \prime}$, and let P be the point with positive $x$-coordinate where the line $y = c$ meets the ellipse. If the difference between the perimeter of triangle $\mathrm { BPF } ^ { \prime}$ and the perimeter of triangle BFA is 4, what is the area of triangle $\mathrm { AFF } ^ { \prime}$? (Given: $0 < a < 5 , c > 0$) [3 points]\\
(1) $3 \sqrt { 6 }$\\
(2) $\frac { 7 \sqrt { 6 } } { 2 }$\\
(3) $4 \sqrt { 6 }$\\
(4) $\frac { 9 \sqrt { 6 } } { 2 }$\\
(5) $5 \sqrt { 6 }$