20. (12 points) Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$, $P$ be a point on $C$, and $O$ be the origin. (1) If $\triangle P O F _ { 2 }$ is an equilateral triangle, find the eccentricity of $C$; (2) If there exists a point $P$ such that $P F _ { 1 } \perp P F _ { 2 }$ and the area of $\triangle F _ { 1 } P F _ { 2 }$ equals 16, find the value of $b$ and the range of values for $a$.
20. (12 points)
Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$, $P$ be a point on $C$, and $O$ be the origin.\\
(1) If $\triangle P O F _ { 2 }$ is an equilateral triangle, find the eccentricity of $C$;\\
(2) If there exists a point $P$ such that $P F _ { 1 } \perp P F _ { 2 }$ and the area of $\triangle F _ { 1 } P F _ { 2 }$ equals 16, find the value of $b$ and the range of values for $a$.