As shown in the figure, there is an ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ with foci at $\mathrm { F } ( c , 0 )$ and $\mathrm { F } ^ { \prime } ( - c , 0 )$. For point P on the ellipse in the second quadrant, let Q be the midpoint of segment $\mathrm { PF } ^ { \prime }$, and let R be the point that divides segment PF internally in the ratio $1 : 3$. When $\angle \mathrm { PQR } = \frac { \pi } { 2 }$, $\overline { \mathrm { QR } } = \sqrt { 5 }$, and $\overline { \mathrm { RF } } = 9$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Here, $a$, $b$, and $c$ are positive numbers.) [4 points]
As shown in the figure, there is an ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ with foci at $\mathrm { F } ( c , 0 )$ and $\mathrm { F } ^ { \prime } ( - c , 0 )$. For point P on the ellipse in the second quadrant, let Q be the midpoint of segment $\mathrm { PF } ^ { \prime }$, and let R be the point that divides segment PF internally in the ratio $1 : 3$. When $\angle \mathrm { PQR } = \frac { \pi } { 2 }$, $\overline { \mathrm { QR } } = \sqrt { 5 }$, and $\overline { \mathrm { RF } } = 9$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Here, $a$, $b$, and $c$ are positive numbers.) [4 points]